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Theorem bj-zfpair2 6477
 Description: Proof of zfpair2 3918 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 6396 . . . . 5 BOUNDED w = x
2 ax-bdeq 6396 . . . . 5 BOUNDED w = y
31, 2ax-bdor 6392 . . . 4 BOUNDED (w = x w = y)
4 ax-pr 3917 . . . 4 zw((w = x w = y) → w z)
53, 4bdbm1.3ii 6461 . . 3 zw(w z ↔ (w = x w = y))
6 dfcleq 2017 . . . . 5 (z = {x, y} ↔ w(w zw {x, y}))
7 vex 2537 . . . . . . . 8 w V
87elpr 3367 . . . . . . 7 (w {x, y} ↔ (w = x w = y))
98bibi2i 216 . . . . . 6 ((w zw {x, y}) ↔ (w z ↔ (w = x w = y)))
109albii 1339 . . . . 5 (w(w zw {x, y}) ↔ w(w z ↔ (w = x w = y)))
116, 10bitri 173 . . . 4 (z = {x, y} ↔ w(w z ↔ (w = x w = y)))
1211exbii 1480 . . 3 (z z = {x, y} ↔ zw(w z ↔ (w = x w = y)))
135, 12mpbir 134 . 2 z z = {x, y}
1413issetri 2541 1 {x, y} V
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∨ wo 616  ∀wal 1226   = wceq 1228  ∃wex 1363   ∈ wcel 1375  Vcvv 2534  {cpr 3350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-pr 3917  ax-bdor 6392  ax-bdeq 6396  ax-bdsep 6455 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-un 2898  df-sn 3355  df-pr 3356 This theorem is referenced by:  bj-prexg  6478
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