MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axun2 Structured version   Visualization version   GIF version

Theorem axun2 6936
Description: A variant of the Axiom of Union ax-un 6934. 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 6934 . 2 𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)
21bm1.3ii 4775 1 𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑧𝑤𝑤𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-11 2032  ax-12 2045  ax-13 2244  ax-sep 4772  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1703
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator