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Axiom ax-un 6824
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 6826 states that the union itself exists. A version with the standard abbreviation for union is uniex2 6827. A version using class notation is uniex 6828.

The union of a class df-uni 4367 should not be confused with the union of two classes df-un 3544. Their relationship is shown in unipr 4379. (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 1977 . . . . . 6 wff 𝑧𝑤
4 vx . . . . . . 7 setvar 𝑥
52, 4wel 1977 . . . . . 6 wff 𝑤𝑥
63, 5wa 382 . . . . 5 wff (𝑧𝑤𝑤𝑥)
76, 2wex 1694 . . . 4 wff 𝑤(𝑧𝑤𝑤𝑥)
8 vy . . . . 5 setvar 𝑦
91, 8wel 1977 . . . 4 wff 𝑧𝑦
107, 9wi 4 . . 3 wff (∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
1110, 1wal 1472 . 2 wff 𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
1211, 8wex 1694 1 wff 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
Colors of variables: wff setvar class
This axiom is referenced by:  zfun  6825  axun2  6826
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