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Theorem bj-zfpair2 15065
Description: Proof of zfpair2 4225 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 14975 . . . . 5 BOUNDED 𝑤 = 𝑥
2 ax-bdeq 14975 . . . . 5 BOUNDED 𝑤 = 𝑦
31, 2ax-bdor 14971 . . . 4 BOUNDED (𝑤 = 𝑥𝑤 = 𝑦)
4 ax-pr 4224 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
53, 4bdbm1.3ii 15046 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
6 dfcleq 2183 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
7 vex 2755 . . . . . . . 8 𝑤 ∈ V
87elpr 3628 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
98bibi2i 227 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
109albii 1481 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
116, 10bitri 184 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
1211exbii 1616 . . 3 (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
135, 12mpbir 146 . 2 𝑧 𝑧 = {𝑥, 𝑦}
1413issetri 2761 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff set class
Syntax hints:  wb 105  wo 709  wal 1362   = wceq 1364  wex 1503  wcel 2160  Vcvv 2752  {cpr 3608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-pr 4224  ax-bdor 14971  ax-bdeq 14975  ax-bdsep 15039
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614
This theorem is referenced by:  bj-prexg  15066
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