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Theorem brcup 24478
 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 4237 . 2
2 brcup.3 . 2
3 df-cup 24410 . 2 Cup (++)
4 brcup.1 . . . 4
5 brcup.2 . . . 4
64, 5opelvv 4735 . . 3
7 brxp 4720 . . 3
86, 2, 7mpbir2an 886 . 2
9 vex 2791 . . . . . . 7
109, 1brco 4852 . . . . . 6
11 ancom 437 . . . . . . . 8
12 vex 2791 . . . . . . . . . . 11
1312, 1brcnv 4864 . . . . . . . . . 10
144, 5, 12br1steq 24130 . . . . . . . . . 10
1513, 14bitri 240 . . . . . . . . 9
1615anbi1i 676 . . . . . . . 8
1711, 16bitri 240 . . . . . . 7
1817exbii 1569 . . . . . 6
1910, 18bitri 240 . . . . 5
20 breq2 4027 . . . . . 6
214, 20ceqsexv 2823 . . . . 5
224epelc 4307 . . . . 5
2319, 21, 223bitri 262 . . . 4
249, 1brco 4852 . . . . 5
25 ancom 437 . . . . . . . 8
2612, 1brcnv 4864 . . . . . . . . . 10
274, 5, 12br2ndeq 24131 . . . . . . . . . 10
2826, 27bitri 240 . . . . . . . . 9
2928anbi1i 676 . . . . . . . 8
3025, 29bitri 240 . . . . . . 7
3130exbii 1569 . . . . . 6
32 breq2 4027 . . . . . . 7
335, 32ceqsexv 2823 . . . . . 6
3431, 33bitri 240 . . . . 5
355epelc 4307 . . . . 5
3624, 34, 353bitri 262 . . . 4
3723, 36orbi12i 507 . . 3
38 brun 4069 . . 3
39 elun 3316 . . 3
4037, 38, 393bitr4ri 269 . 2
411, 2, 3, 8, 40brtxpsd3 24436 1 Cup
 Colors of variables: wff set class Syntax hints:   wb 176   wo 357   wa 358  wex 1528   wceq 1623   wcel 1684  cvv 2788   cun 3150  cop 3643   class class class wbr 4023   cep 4303   cxp 4687  ccnv 4688   ccom 4693  c1st 6120  c2nd 6121  Cupccup 24389 This theorem is referenced by:  brsuccf  24480 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-cup 24410
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