Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brsuccf Structured version   Unicode version

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

Proof of Theorem brsuccf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-succf 25716 . . 3 Succ Cup Singleton
21breqi 4218 . 2 Succ Cup Singleton
3 brsuccf.1 . . . . 5
4 brsuccf.2 . . . . 5
53, 4brco 5043 . . . 4 Cup Singleton Singleton Cup
63brtxp2 25726 . . . . . . . . 9 Singleton Singleton
7 3anass 940 . . . . . . . . . . 11 Singleton Singleton
8 vex 2959 . . . . . . . . . . . . . . 15
98ideq 5025 . . . . . . . . . . . . . 14
10 eqcom 2438 . . . . . . . . . . . . . 14
119, 10bitri 241 . . . . . . . . . . . . 13
12 vex 2959 . . . . . . . . . . . . . 14
133, 12brsingle 25762 . . . . . . . . . . . . 13 Singleton
1411, 13anbi12i 679 . . . . . . . . . . . 12 Singleton
1514anbi2i 676 . . . . . . . . . . 11 Singleton
167, 15bitri 241 . . . . . . . . . 10 Singleton
17162exbii 1593 . . . . . . . . 9 Singleton
186, 17bitri 241 . . . . . . . 8 Singleton
1918anbi1i 677 . . . . . . 7 Singleton Cup Cup
20 19.41vv 1925 . . . . . . 7 Cup Cup
2119, 20bitr4i 244 . . . . . 6 Singleton Cup Cup
22 anass 631 . . . . . . 7 Cup Cup
23222exbii 1593 . . . . . 6 Cup Cup
2421, 23bitri 241 . . . . 5 Singleton Cup Cup
2524exbii 1592 . . . 4 Singleton Cup Cup
26 excom 1756 . . . . 5 Cup Cup
27 excom 1756 . . . . . . 7 Cup Cup
28 opex 4427 . . . . . . . . . 10
29 breq1 4215 . . . . . . . . . . . 12 Cup Cup
308, 12, 4brcup 25784 . . . . . . . . . . . 12 Cup
3129, 30syl6bb 253 . . . . . . . . . . 11 Cup
3231anbi2d 685 . . . . . . . . . 10 Cup
3328, 32ceqsexv 2991 . . . . . . . . 9 Cup
34 df-3an 938 . . . . . . . . 9
3533, 34bitr4i 244 . . . . . . . 8 Cup
3635exbii 1592 . . . . . . 7 Cup
3727, 36bitri 241 . . . . . 6 Cup
3837exbii 1592 . . . . 5 Cup
39 snex 4405 . . . . . 6
40 uneq1 3494 . . . . . . 7
4140eqeq2d 2447 . . . . . 6
42 uneq2 3495 . . . . . . 7
4342eqeq2d 2447 . . . . . 6
443, 39, 41, 43ceqsex2v 2993 . . . . 5
4526, 38, 443bitri 263 . . . 4 Cup
465, 25, 453bitri 263 . . 3 Cup Singleton
47 df-suc 4587 . . . 4
4847eqeq2i 2446 . . 3
4946, 48bitr4i 244 . 2 Cup Singleton
502, 49bitri 241 1 Succ
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   w3a 936  wex 1550   wceq 1652   wcel 1725  cvv 2956   cun 3318  csn 3814  cop 3817   class class class wbr 4212   cid 4493   csuc 4583   ccom 4882   ctxp 25674  Singletoncsingle 25682  Cupccup 25690  Succcsuccf 25692 This theorem is referenced by:  dfrdg4  25795 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-eprel 4494  df-id 4498  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-symdif 25663  df-txp 25698  df-singleton 25706  df-cup 25713  df-succf 25716
 Copyright terms: Public domain W3C validator