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Theorem brsuccf 25038
 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 24971 . . 3 Succ Cup Singleton
21breqi 4110 . 2 Succ Cup Singleton
3 brsuccf.1 . . . . 5
4 brsuccf.2 . . . . 5
53, 4brco 4934 . . . 4 Cup Singleton Singleton Cup
63brtxp2 24979 . . . . . . . . 9 Singleton Singleton
7 3anass 938 . . . . . . . . . . 11 Singleton Singleton
8 vex 2867 . . . . . . . . . . . . . . 15
98ideq 4918 . . . . . . . . . . . . . 14
10 eqcom 2360 . . . . . . . . . . . . . 14
119, 10bitri 240 . . . . . . . . . . . . 13
12 vex 2867 . . . . . . . . . . . . . 14
133, 12brsingle 25014 . . . . . . . . . . . . 13 Singleton
1411, 13anbi12i 678 . . . . . . . . . . . 12 Singleton
1514anbi2i 675 . . . . . . . . . . 11 Singleton
167, 15bitri 240 . . . . . . . . . 10 Singleton
17162exbii 1583 . . . . . . . . 9 Singleton
186, 17bitri 240 . . . . . . . 8 Singleton
1918anbi1i 676 . . . . . . 7 Singleton Cup Cup
20 19.41vv 1907 . . . . . . 7 Cup Cup
2119, 20bitr4i 243 . . . . . 6 Singleton Cup Cup
22 anass 630 . . . . . . 7 Cup Cup
23222exbii 1583 . . . . . 6 Cup Cup
2421, 23bitri 240 . . . . 5 Singleton Cup Cup
2524exbii 1582 . . . 4 Singleton Cup Cup
26 excom 1741 . . . . 5 Cup Cup
27 excom 1741 . . . . . . 7 Cup Cup
28 opex 4319 . . . . . . . . . 10
29 breq1 4107 . . . . . . . . . . . 12 Cup Cup
308, 12, 4brcup 25036 . . . . . . . . . . . 12 Cup
3129, 30syl6bb 252 . . . . . . . . . . 11 Cup
3231anbi2d 684 . . . . . . . . . 10 Cup
3328, 32ceqsexv 2899 . . . . . . . . 9 Cup
34 df-3an 936 . . . . . . . . 9
3533, 34bitr4i 243 . . . . . . . 8 Cup
3635exbii 1582 . . . . . . 7 Cup
3727, 36bitri 240 . . . . . 6 Cup
3837exbii 1582 . . . . 5 Cup
39 snex 4297 . . . . . 6
40 uneq1 3398 . . . . . . 7
4140eqeq2d 2369 . . . . . 6
42 uneq2 3399 . . . . . . 7
4342eqeq2d 2369 . . . . . 6
443, 39, 41, 43ceqsex2v 2901 . . . . 5
4526, 38, 443bitri 262 . . . 4 Cup
465, 25, 453bitri 262 . . 3 Cup Singleton
47 df-suc 4480 . . . 4
4847eqeq2i 2368 . . 3
4946, 48bitr4i 243 . 2 Cup Singleton
502, 49bitri 240 1 Succ
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   w3a 934  wex 1541   wceq 1642   wcel 1710  cvv 2864   cun 3226  csn 3716  cop 3719   class class class wbr 4104   cid 4386   csuc 4476   ccom 4775   ctxp 24931  Singletoncsingle 24939  Cupccup 24947  Succcsuccf 24949 This theorem is referenced by:  dfrdg4  25047 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-eprel 4387  df-id 4391  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fo 5343  df-fv 5345  df-1st 6209  df-2nd 6210  df-symdif 24920  df-txp 24953  df-singleton 24961  df-cup 24968  df-succf 24971
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