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Theorem axun2 4262
Description: A variant of the Axiom of Union ax-un 4260. 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 4260 . 2 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
21bm1.3ii 3960 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wal 1287  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-sep 3957  ax-un 4260
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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