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Axiom ax-un 6659
Description: Axiom of Union. An axiom of Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the union of a given set 𝑥 i.e. the collection of all members of the members of 𝑥. The variant axun2 6661 states that the union itself exists. A version with the standard abbreviation for union is uniex2 6662. A version using class notation is uniex 6663.

The union of a class df-uni 4229 should not be confused with the union of two classes df-un 3431. Their relationship is shown in unipr 4241. (Contributed by NM, 23-Dec-1993.)

Assertion
Ref Expression
ax-un 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Detailed syntax breakdown of Axiom ax-un
StepHypRef Expression
1 vz . . . . . . 7 setvar 𝑧
2 vw . . . . . . 7 setvar 𝑤
31, 2wel 1938 . . . . . 6 wff 𝑧𝑤
4 vx . . . . . . 7 setvar 𝑥
52, 4wel 1938 . . . . . 6 wff 𝑤𝑥
63, 5wa 378 . . . . 5 wff (𝑧𝑤𝑤𝑥)
76, 2wex 1692 . . . 4 wff 𝑤(𝑧𝑤𝑤𝑥)
8 vy . . . . 5 setvar 𝑦
91, 8wel 1938 . . . 4 wff 𝑧𝑦
107, 9wi 4 . . 3 wff (∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
1110, 1wal 1466 . 2 wff 𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
1211, 8wex 1692 1 wff 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
This axiom is referenced by:  zfun  6660  axun2  6661
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