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Theorem bj-zfpair2 9365
Description: Proof of zfpair2 3936 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 9275 . . . . 5 BOUNDED w = x
2 ax-bdeq 9275 . . . . 5 BOUNDED w = y
31, 2ax-bdor 9271 . . . 4 BOUNDED (w = x w = y)
4 ax-pr 3935 . . . 4 zw((w = x w = y) → w z)
53, 4bdbm1.3ii 9346 . . 3 zw(w z ↔ (w = x w = y))
6 dfcleq 2031 . . . . 5 (z = {x, y} ↔ w(w zw {x, y}))
7 vex 2554 . . . . . . . 8 w V
87elpr 3385 . . . . . . 7 (w {x, y} ↔ (w = x w = y))
98bibi2i 216 . . . . . 6 ((w zw {x, y}) ↔ (w z ↔ (w = x w = y)))
109albii 1356 . . . . 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 1493 . . 3 (z z = {x, y} ↔ zw(w z ↔ (w = x w = y)))
135, 12mpbir 134 . 2 z z = {x, y}
1413issetri 2558 1 {x, y} V
Colors of variables: wff set class
Syntax hints:  wb 98   wo 628  wal 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  {cpr 3368
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-pr 3935  ax-bdor 9271  ax-bdeq 9275  ax-bdsep 9339
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374
This theorem is referenced by:  bj-prexg  9366
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