Theorem brsuccf 27972

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	|- A e. _V
brsuccf.2	|- B e. _V

Assertion

Ref	Expression
brsuccf	|- ( ASucc B <-> B = suc A )

Proof of Theorem brsuccf

Dummy variables a b x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-succf 27902	. . 3 |- Succ = (Cup o. ( _I (x) Singleton ) )
2	1	breqi 4298	. 2 |- ( ASucc B <-> A (Cup o. ( _I (x) Singleton ) ) B )
3		brsuccf.1	. . . . 5 |- A e. _V
4		brsuccf.2	. . . . 5 |- B e. _V
5	3, 4	brco 5010	. . . 4 |- ( A (Cup o. ( _I (x) Singleton ) ) B <-> E. x ( A ( _I (x) Singleton ) x /\ xCup B ) )
6	3	brtxp2 27912	. . . . . . . . 9 |- ( A ( _I (x) Singleton ) x <-> E. a E. b ( x = <. a , b >. /\ A _I a /\ ASingleton b ) )
7		3anass 969	. . . . . . . . . . 11 |- ( ( x = <. a , b >. /\ A _I a /\ ASingleton b ) <-> ( x = <. a , b >. /\ ( A _I a /\ ASingleton b ) ) )
8		vex 2975	. . . . . . . . . . . . . . 15 |- a e. _V
9	8	ideq 4992	. . . . . . . . . . . . . 14 |- ( A _I a <-> A = a )
10		eqcom 2445	. . . . . . . . . . . . . 14 |- ( A = a <-> a = A )
11	9, 10	bitri 249	. . . . . . . . . . . . 13 |- ( A _I a <-> a = A )
12		vex 2975	. . . . . . . . . . . . . 14 |- b e. _V
13	3, 12	brsingle 27948	. . . . . . . . . . . . 13 |- ( ASingleton b <-> b = { A } )
14	11, 13	anbi12i 697	. . . . . . . . . . . 12 |- ( ( A _I a /\ ASingleton b ) <-> ( a = A /\ b = { A } ) )
15	14	anbi2i 694	. . . . . . . . . . 11 |- ( ( x = <. a , b >. /\ ( A _I a /\ ASingleton b ) ) <-> ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) )
16	7, 15	bitri 249	. . . . . . . . . 10 |- ( ( x = <. a , b >. /\ A _I a /\ ASingleton b ) <-> ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) )
17	16	2exbii 1635	. . . . . . . . 9 |- ( E. a E. b ( x = <. a , b >. /\ A _I a /\ ASingleton b ) <-> E. a E. b ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) )
18	6, 17	bitri 249	. . . . . . . 8 |- ( A ( _I (x) Singleton ) x <-> E. a E. b ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) )
19	18	anbi1i 695	. . . . . . 7 |- ( ( A ( _I (x) Singleton ) x /\ xCup B ) <-> ( E. a E. b ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) /\ xCup B ) )
20		19.41vv 1921	. . . . . . 7 |- ( E. a E. b ( ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) /\ xCup B ) <-> ( E. a E. b ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) /\ xCup B ) )
21	19, 20	bitr4i 252	. . . . . 6 |- ( ( A ( _I (x) Singleton ) x /\ xCup B ) <-> E. a E. b ( ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) /\ xCup B ) )
22		anass 649	. . . . . . 7 |- ( ( ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) /\ xCup B ) <-> ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) )
23	22	2exbii 1635	. . . . . 6 |- ( E. a E. b ( ( x = <. a , b >. /\ ( a = A /\ b = { A } ) ) /\ xCup B ) <-> E. a E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) )
24	21, 23	bitri 249	. . . . 5 |- ( ( A ( _I (x) Singleton ) x /\ xCup B ) <-> E. a E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) )
25	24	exbii 1634	. . . 4 |- ( E. x ( A ( _I (x) Singleton ) x /\ xCup B ) <-> E. x E. a E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) )
26		excom 1787	. . . . 5 |- ( E. x E. a E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> E. a E. x E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) )
27		excom 1787	. . . . . . 7 |- ( E. x E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> E. b E. x ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) )
28		opex 4556	. . . . . . . . . 10 |- <. a , b >. e. _V
29		breq1 4295	. . . . . . . . . . . 12 |- ( x = <. a , b >. -> ( xCup B <-> <. a , b >.Cup B ) )
30	8, 12, 4	brcup 27970	. . . . . . . . . . . 12 |- ( <. a , b >.Cup B <-> B = ( a u. b ) )
31	29, 30	syl6bb 261	. . . . . . . . . . 11 |- ( x = <. a , b >. -> ( xCup B <-> B = ( a u. b ) ) )
32	31	anbi2d 703	. . . . . . . . . 10 |- ( x = <. a , b >. -> ( ( ( a = A /\ b = { A } ) /\ xCup B ) <-> ( ( a = A /\ b = { A } ) /\ B = ( a u. b ) ) ) )
33	28, 32	ceqsexv 3009	. . . . . . . . 9 |- ( E. x ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> ( ( a = A /\ b = { A } ) /\ B = ( a u. b ) ) )
34		df-3an 967	. . . . . . . . 9 |- ( ( a = A /\ b = { A } /\ B = ( a u. b ) ) <-> ( ( a = A /\ b = { A } ) /\ B = ( a u. b ) ) )
35	33, 34	bitr4i 252	. . . . . . . 8 |- ( E. x ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> ( a = A /\ b = { A } /\ B = ( a u. b ) ) )
36	35	exbii 1634	. . . . . . 7 |- ( E. b E. x ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> E. b ( a = A /\ b = { A } /\ B = ( a u. b ) ) )
37	27, 36	bitri 249	. . . . . 6 |- ( E. x E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> E. b ( a = A /\ b = { A } /\ B = ( a u. b ) ) )
38	37	exbii 1634	. . . . 5 |- ( E. a E. x E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> E. a E. b ( a = A /\ b = { A } /\ B = ( a u. b ) ) )
39		snex 4533	. . . . . 6 |- { A } e. _V
40		uneq1 3503	. . . . . . 7 |- ( a = A -> ( a u. b ) = ( A u. b ) )
41	40	eqeq2d 2454	. . . . . 6 |- ( a = A -> ( B = ( a u. b ) <-> B = ( A u. b ) ) )
42		uneq2 3504	. . . . . . 7 |- ( b = { A } -> ( A u. b ) = ( A u. { A } ) )
43	42	eqeq2d 2454	. . . . . 6 |- ( b = { A } -> ( B = ( A u. b ) <-> B = ( A u. { A } ) ) )
44	3, 39, 41, 43	ceqsex2v 3011	. . . . 5 |- ( E. a E. b ( a = A /\ b = { A } /\ B = ( a u. b ) ) <-> B = ( A u. { A } ) )
45	26, 38, 44	3bitri 271	. . . 4 |- ( E. x E. a E. b ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> B = ( A u. { A } ) )
46	5, 25, 45	3bitri 271	. . 3 |- ( A (Cup o. ( _I (x) Singleton ) ) B <-> B = ( A u. { A } ) )
47		df-suc 4725	. . . 4 |- suc A = ( A u. { A } )
48	47	eqeq2i 2453	. . 3 |- ( B = suc A <-> B = ( A u. { A } ) )
49	46, 48	bitr4i 252	. 2 |- ( A (Cup o. ( _I (x) Singleton ) ) B <-> B = suc A )
50	2, 49	bitri 249	1 |- ( ASucc B <-> B = suc A )

Syntax hints:   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   _Vcvv 2972   u. cun 3326   {csn 3877   <.cop 3883   class class class wbr 4292   _I cid 4631   suc csuc 4721   o. ccom 4844   (x) ctxp 27860  Singletoncsingle 27868  Cupccup 27876  Succcsuccf 27878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-eprel 4632  df-id 4636  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-1st 6577  df-2nd 6578  df-symdif 27849  df-txp 27884  df-singleton 27892  df-cup 27899  df-succf 27902

This theorem is referenced by:  dfrdg4  27981