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Theorem brcup 25786
 Description: Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcup.1
brcup.2
brcup.3
Assertion
Ref Expression
brcup Cup

Proof of Theorem brcup
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4429 . 2
2 brcup.3 . 2
3 df-cup 25715 . 2 Cup (++)
4 brcup.1 . . . 4
5 brcup.2 . . . 4
64, 5opelvv 4926 . . 3
7 brxp 4911 . . 3
86, 2, 7mpbir2an 888 . 2
9 epel 4499 . . . . . . 7
10 vex 2961 . . . . . . . . 9
1110, 1brcnv 5057 . . . . . . . 8
124, 5, 10br1steq 25400 . . . . . . . 8
1311, 12bitri 242 . . . . . . 7
149, 13anbi12ci 681 . . . . . 6
1514exbii 1593 . . . . 5
16 vex 2961 . . . . . 6
1716, 1brco 5045 . . . . 5
184clel3 3076 . . . . 5
1915, 17, 183bitr4i 270 . . . 4
2010, 1brcnv 5057 . . . . . . . 8
214, 5, 10br2ndeq 25401 . . . . . . . 8
2220, 21bitri 242 . . . . . . 7
239, 22anbi12ci 681 . . . . . 6
2423exbii 1593 . . . . 5
2516, 1brco 5045 . . . . 5
265clel3 3076 . . . . 5
2724, 25, 263bitr4i 270 . . . 4
2819, 27orbi12i 509 . . 3
29 brun 4260 . . 3
30 elun 3490 . . 3
3128, 29, 303bitr4ri 271 . 2
321, 2, 3, 8, 31brtxpsd3 25743 1 Cup
 Colors of variables: wff set class Syntax hints:   wb 178   wo 359   wa 360  wex 1551   wceq 1653   wcel 1726  cvv 2958   cun 3320  cop 3819   class class class wbr 4214   cep 4494   cxp 4878  ccnv 4879   ccom 4884  c1st 6349  c2nd 6350  Cupccup 25692 This theorem is referenced by:  brsuccf  25788 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-eprel 4496  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-1st 6351  df-2nd 6352  df-symdif 25665  df-txp 25700  df-cup 25715
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