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

Theorem bj-axsn 37014
Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 37015). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-axsn ({𝑥} ∈ V ↔ ∃𝑦𝑧(𝑧𝑦𝑧 = 𝑥))
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem bj-axsn
StepHypRef Expression
1 velsn 4646 . 2 (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥)
21bj-clex 37013 1 ({𝑥} ∈ V ↔ ∃𝑦𝑧(𝑧𝑦𝑧 = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1534  wex 1775  wcel 2105  Vcvv 3477  {csn 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-sn 4631
This theorem is referenced by:  bj-snexg  37016
  Copyright terms: Public domain W3C validator