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Theorem axun2 7441
 Description: A variant of the Axiom of Union ax-un 7439. For any set 𝑥, there exists a set 𝑦 whose members are exactly the members of the members of 𝑥 i.e. the union of 𝑥. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
axun2 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Proof of Theorem axun2
StepHypRef Expression
1 ax-un 7439 . 2 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
21bm1.3ii 5182 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398  ∀wal 1535  ∃wex 1780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2124  ax-sep 5179  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781 This theorem is referenced by: (None)
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