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Theorem bj-zfpair2 13905
Description: Proof of zfpair2 4193 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 13815 . . . . 5 BOUNDED 𝑤 = 𝑥
2 ax-bdeq 13815 . . . . 5 BOUNDED 𝑤 = 𝑦
31, 2ax-bdor 13811 . . . 4 BOUNDED (𝑤 = 𝑥𝑤 = 𝑦)
4 ax-pr 4192 . . . 4 𝑧𝑤((𝑤 = 𝑥𝑤 = 𝑦) → 𝑤𝑧)
53, 4bdbm1.3ii 13886 . . 3 𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦))
6 dfcleq 2164 . . . . 5 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}))
7 vex 2733 . . . . . . . 8 𝑤 ∈ V
87elpr 3602 . . . . . . 7 (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥𝑤 = 𝑦))
98bibi2i 226 . . . . . 6 ((𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
109albii 1463 . . . . 5 (∀𝑤(𝑤𝑧𝑤 ∈ {𝑥, 𝑦}) ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
116, 10bitri 183 . . . 4 (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
1211exbii 1598 . . 3 (∃𝑧 𝑧 = {𝑥, 𝑦} ↔ ∃𝑧𝑤(𝑤𝑧 ↔ (𝑤 = 𝑥𝑤 = 𝑦)))
135, 12mpbir 145 . 2 𝑧 𝑧 = {𝑥, 𝑦}
1413issetri 2739 1 {𝑥, 𝑦} ∈ V
Colors of variables: wff set class
Syntax hints:  wb 104  wo 703  wal 1346   = wceq 1348  wex 1485  wcel 2141  Vcvv 2730  {cpr 3582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-pr 4192  ax-bdor 13811  ax-bdeq 13815  ax-bdsep 13879
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588
This theorem is referenced by:  bj-prexg  13906
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