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Theorem difopab 5139
Description: The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
difopab
Distinct variable group:   ,

Proof of Theorem difopab
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5134 . . 3
2 reldif 5127 . . 3
31, 2ax-mp 5 . 2
4 relopab 5134 . 2
5 sbcan 3370 . . . 4
6 sbcan 3370 . . . . 5
76sbcbii 3387 . . . 4
8 opelopabsb 4762 . . . . 5
9 vex 3112 . . . . . . 7
10 sbcng 3368 . . . . . . 7
119, 10ax-mp 5 . . . . . 6
12 vex 3112 . . . . . . . 8
13 sbcng 3368 . . . . . . . 8
1412, 13ax-mp 5 . . . . . . 7
1514sbcbii 3387 . . . . . 6
16 opelopabsb 4762 . . . . . . 7
1716notbii 296 . . . . . 6
1811, 15, 173bitr4ri 278 . . . . 5
198, 18anbi12i 697 . . . 4
205, 7, 193bitr4ri 278 . . 3
21 eldif 3485 . . 3
22 opelopabsb 4762 . . 3
2320, 21, 223bitr4i 277 . 2
243, 4, 23eqrelriiv 5102 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  [.wsbc 3327  \cdif 3472  <.cop 4035  {copab 4509  Relwrel 5009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-xp 5010  df-rel 5011
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