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Theorem brcup 29166
 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 4711 . 2
2 brcup.3 . 2
3 df-cup 29095 . 2 Cup (++)
4 brcup.1 . . . 4
5 brcup.2 . . . 4
64, 5opelvv 5045 . . 3
7 brxp 5029 . . 3
86, 2, 7mpbir2an 918 . 2
9 epel 4794 . . . . . . 7
10 vex 3116 . . . . . . . . 9
1110, 1brcnv 5183 . . . . . . . 8
124, 5, 10br1steq 28781 . . . . . . . 8
1311, 12bitri 249 . . . . . . 7
149, 13anbi12ci 698 . . . . . 6
1514exbii 1644 . . . . 5
16 vex 3116 . . . . . 6
1716, 1brco 5171 . . . . 5
184clel3 3242 . . . . 5
1915, 17, 183bitr4i 277 . . . 4
2010, 1brcnv 5183 . . . . . . . 8
214, 5, 10br2ndeq 28782 . . . . . . . 8
2220, 21bitri 249 . . . . . . 7
239, 22anbi12ci 698 . . . . . 6
2423exbii 1644 . . . . 5
2516, 1brco 5171 . . . . 5
265clel3 3242 . . . . 5
2724, 25, 263bitr4i 277 . . . 4
2819, 27orbi12i 521 . . 3
29 brun 4495 . . 3
30 elun 3645 . . 3
3128, 29, 303bitr4ri 278 . 2
321, 2, 3, 8, 31brtxpsd3 29123 1 Cup
 Colors of variables: wff setvar class Syntax hints:   wb 184   wo 368   wa 369   wceq 1379  wex 1596   wcel 1767  cvv 3113   cun 3474  cop 4033   class class class wbr 4447   cep 4789   cxp 4997  ccnv 4998   ccom 5003  c1st 6779  c2nd 6780  Cupccup 29072 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-eprel 4791  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-1st 6781  df-2nd 6782  df-symdif 29045  df-txp 29080  df-cup 29095 This theorem is referenced by:  brsuccf  29168
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