Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-axbun Structured version   Visualization version   GIF version

Theorem bj-axbun 36423
Description: Two ways of stating the axiom of binary union (which is the universal closure of either side, see ax-bj-bun 36424). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axbun ((𝑥𝑦) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦)))
Distinct variable groups:   𝑥,𝑧,𝑡   𝑦,𝑧,𝑡

Proof of Theorem bj-axbun
StepHypRef Expression
1 elun 4143 . 2 (𝑡 ∈ (𝑥𝑦) ↔ (𝑡𝑥𝑡𝑦))
21bj-clex 36418 1 ((𝑥𝑦) ∈ V ↔ ∃𝑧𝑡(𝑡𝑧 ↔ (𝑡𝑥𝑡𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 844  wal 1531  wex 1773  wcel 2098  Vcvv 3468  cun 3941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-un 3948
This theorem is referenced by:  bj-unexg  36425
  Copyright terms: Public domain W3C validator