Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-zfpair2 GIF version

Theorem bj-zfpair2 12689
Description: Proof of zfpair2 4070 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-zfpair2 {𝑥, 𝑦} ∈ V

Proof of Theorem bj-zfpair2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-bdeq 12599 . . . . 5 BOUNDED 𝑤 = 𝑥
2 ax-bdeq 12599 . . . . 5 BOUNDED 𝑤 = 𝑦
31, 2ax-bdor 12595 . . . 4 BOUNDED (𝑤 = 𝑥𝑤 = 𝑦)
4 ax-pr 4069 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
53, 4bdbm1.3ii 12670 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
6 dfcleq 2094 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
7 vex 2644 . . . . . . . 8 𝑤 ∈ V
87elpr 3495 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
98bibi2i 226 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
109albii 1414 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
116, 10bitri 183 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
1211exbii 1552 . . 3 (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
135, 12mpbir 145 . 2 𝑧 𝑧 = {𝑥, 𝑦}
1413issetri 2650 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff set class
Syntax hints:  wb 104  wo 670  wal 1297   = wceq 1299  wex 1436  wcel 1448  Vcvv 2641  {cpr 3475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-pr 4069  ax-bdor 12595  ax-bdeq 12599  ax-bdsep 12663
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481
This theorem is referenced by:  bj-prexg  12690
  Copyright terms: Public domain W3C validator