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Theorem axun2 4172
 Description: A variant of the Axiom of Union ax-un 4170. 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 4170 . 2 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
21bm1.3ii 3878 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-sep 3875  ax-un 4170 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
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