No.

The antimagic field suppresses all of the effects of the haste spell, but it doesn't end the spell. While a creature under the effects of haste is inside an antimagic field, none of the effects (positive or negative) of the spell apply, but the spell itself still exists and hasn't ended.

Similarly, if a haste spell does end while the affected creature is inside an antimagic field, then there will be no "wave of lethargy"; that is also an effect of the spell, and it is also suppressed.

Contrast the wording of antimagic shell (the spell whose effects define those of an antimagic field) with that of dispel magic (PHB p., 234, emphasis mine):

Choose one creature, object, or magical effect within range. Any spell of 3rd level or lower on the target ends.

Dispel magic explicitly ends spells; if antimagic field and similar effects also did so, they would explicitly say so.

Antimagic Field will prevail

We can look at this from two points of view, both from the rules and from a purely RP perspective. Let's check the rules first.

The description of Antimagic Field states that

Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it. A slot expended to cast a suppressed spell is consumed. While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.

and that

Spells and magical effects such as dispel magic have no effect on the Sphere. Likewise, the spheres created by different antimagic field spells don't nullify each other

However, the Globe of Invulnerability can be dispelled or suppressed, as it specifies that only what is inside it is protected, and not the globe itself. Therefore it stands to reason that when the two spells meet, Antimagic Field will suppress the Globe of Invulnerability itself, which will allow the field to affect creatures or objects within the globe.

If we look at it from a RP point of view, we can once again turn to Antimagic Field's description, which says of the area of the field:

This area is divorced from the magical energy that suffuses the multiverse.

We could then say that this area lacks the required property to sustain a spell, which would suppress the globe. This isn't a case of a spell being stronger than another, such as light spells vs darkness spells. This is an instance of a spell removing the necessary energy for another spell to sustain itself, thereby making it fail.