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Theorem difopab 4966
 Description: The difference of two ordered-pair abstractions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
difopab
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem difopab
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4960 . . 3
2 reldif 4953 . . 3
31, 2ax-mp 5 . 2
4 relopab 4960 . 2
5 sbcan 3310 . . . 4
6 sbcan 3310 . . . . 5
76sbcbii 3323 . . . 4
8 opelopabsb 4711 . . . . 5
9 vex 3048 . . . . . . 7
10 sbcng 3308 . . . . . . 7
119, 10ax-mp 5 . . . . . 6
12 vex 3048 . . . . . . . 8
13 sbcng 3308 . . . . . . . 8
1412, 13ax-mp 5 . . . . . . 7
1514sbcbii 3323 . . . . . 6
16 opelopabsb 4711 . . . . . . 7
1716notbii 298 . . . . . 6
1811, 15, 173bitr4ri 282 . . . . 5
198, 18anbi12i 703 . . . 4
205, 7, 193bitr4ri 282 . . 3
21 eldif 3414 . . 3
22 opelopabsb 4711 . . 3
2320, 21, 223bitr4i 281 . 2
243, 4, 23eqrelriiv 4929 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 188   wa 371   wceq 1444   wcel 1887  cvv 3045  wsbc 3267   cdif 3401  cop 3974  copab 4460   wrel 4839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-opab 4462  df-xp 4840  df-rel 4841 This theorem is referenced by: (None)
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