Theorem brsuccf 29518

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 29448	. . 3 |- Succ = (Cup o. ( _I (x) Singleton ) )
2	1	breqi 4459	. 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 5179	. . . 4 |- ( A (Cup o. ( _I (x) Singleton ) ) B <-> E. x ( A ( _I (x) Singleton ) x /\ xCup B ) )
6	3	brtxp2 29458	. . . . . . . . 9 |- ( A ( _I (x) Singleton ) x <-> E. a E. b ( x = <. a , b >. /\ A _I a /\ ASingleton b ) )
7		3anass 977	. . . . . . . . . . 11 |- ( ( x = <. a , b >. /\ A _I a /\ ASingleton b ) <-> ( x = <. a , b >. /\ ( A _I a /\ ASingleton b ) ) )
8		vex 3121	. . . . . . . . . . . . . . 15 |- a e. _V
9	8	ideq 5161	. . . . . . . . . . . . . 14 |- ( A _I a <-> A = a )
10		eqcom 2476	. . . . . . . . . . . . . 14 |- ( A = a <-> a = A )
11	9, 10	bitri 249	. . . . . . . . . . . . 13 |- ( A _I a <-> a = A )
12		vex 3121	. . . . . . . . . . . . . 14 |- b e. _V
13	3, 12	brsingle 29494	. . . . . . . . . . . . 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 1645	. . . . . . . . 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 1946	. . . . . . 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 1645	. . . . . 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 1644	. . . 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 1798	. . . . 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 1798	. . . . . . 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 4717	. . . . . . . . . 10 |- <. a , b >. e. _V
29		breq1 4456	. . . . . . . . . . . 12 |- ( x = <. a , b >. -> ( xCup B <-> <. a , b >.Cup B ) )
30	8, 12, 4	brcup 29516	. . . . . . . . . . . 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 3155	. . . . . . . . 9 |- ( E. x ( x = <. a , b >. /\ ( ( a = A /\ b = { A } ) /\ xCup B ) ) <-> ( ( a = A /\ b = { A } ) /\ B = ( a u. b ) ) )
34		df-3an 975	. . . . . . . . 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 1644	. . . . . . 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 1644	. . . . 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 4694	. . . . . 6 |- { A } e. _V
40		uneq1 3656	. . . . . . 7 |- ( a = A -> ( a u. b ) = ( A u. b ) )
41	40	eqeq2d 2481	. . . . . 6 |- ( a = A -> ( B = ( a u. b ) <-> B = ( A u. b ) ) )
42		uneq2 3657	. . . . . . 7 |- ( b = { A } -> ( A u. b ) = ( A u. { A } ) )
43	42	eqeq2d 2481	. . . . . 6 |- ( b = { A } -> ( B = ( A u. b ) <-> B = ( A u. { A } ) ) )
44	3, 39, 41, 43	ceqsex2v 3157	. . . . 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 4890	. . . 4 |- suc A = ( A u. { A } )
48	47	eqeq2i 2485	. . 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 973   = wceq 1379   E.wex 1596   e. wcel 1767   _Vcvv 3118   u. cun 3479   {csn 4033   <.cop 4039   class class class wbr 4453   _I cid 4796   suc csuc 4886   o. ccom 5009   (x) ctxp 29406  Singletoncsingle 29414  Cupccup 29422  Succcsuccf 29424

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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587

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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-eprel 4797  df-id 4801  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-1st 6795  df-2nd 6796  df-symdif 29395  df-txp 29430  df-singleton 29438  df-cup 29445  df-succf 29448

This theorem is referenced by:  dfrdg4  29527