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Theorem axun2 7097
Description: A variant of the Axiom of Union ax-un 7095. 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 7095 . 2 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
21bm1.3ii 4915 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wal 1628  wex 1851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-11 2189  ax-12 2202  ax-13 2407  ax-sep 4912  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857
This theorem is referenced by: (None)
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