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Theorem difopab 4998
 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 4993 . . 3
2 reldif 4986 . . 3
31, 2ax-mp 8 . 2
4 relopab 4993 . 2
5 sbcan 3195 . . . 4
6 sbcan 3195 . . . . 5
76sbcbii 3208 . . . 4
8 opelopabsb 4457 . . . . 5
9 vex 2951 . . . . . . 7
10 sbcng 3193 . . . . . . 7
119, 10ax-mp 8 . . . . . 6
12 vex 2951 . . . . . . . 8
13 sbcng 3193 . . . . . . . 8
1412, 13ax-mp 8 . . . . . . 7
1514sbcbii 3208 . . . . . 6
16 opelopabsb 4457 . . . . . . 7
1716notbii 288 . . . . . 6
1811, 15, 173bitr4ri 270 . . . . 5
198, 18anbi12i 679 . . . 4
205, 7, 193bitr4ri 270 . . 3
21 eldif 3322 . . 3
22 opelopabsb 4457 . . 3
2320, 21, 223bitr4i 269 . 2
243, 4, 23eqrelriiv 4962 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2948  wsbc 3153   cdif 3309  cop 3809  copab 4257   wrel 4875 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-rel 4877
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