Theorem sucdom2 7707

Description: Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)

Assertion

Ref	Expression
sucdom2	|- ( A ~< B -> suc A ~<_ B )

Proof of Theorem sucdom2

Dummy variables w f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sdomdom 7536	. . 3 |- ( A ~< B -> A ~<_ B )
2		brdomi 7520	. . 3 |- ( A ~<_ B -> E. f f : A -1-1-> B )
3	1, 2	syl 16	. 2 |- ( A ~< B -> E. f f : A -1-1-> B )
4		relsdom 7516	. . . . . . 7 |- Rel ~<
5	4	brrelexi 5029	. . . . . 6 |- ( A ~< B -> A e. _V )
6	5	adantr 463	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A e. _V )
7		vex 3109	. . . . . . 7 |- f e. _V
8	7	rnex 6707	. . . . . 6 |- ran f e. _V
9	8	a1i 11	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f e. _V )
10		f1f1orn 5809	. . . . . . 7 |- ( f : A -1-1-> B -> f : A -1-1-onto-> ran f )
11	10	adantl 464	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> f : A -1-1-onto-> ran f )
12		f1of1 5797	. . . . . 6 |- ( f : A -1-1-onto-> ran f -> f : A -1-1-> ran f )
13	11, 12	syl 16	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> f : A -1-1-> ran f )
14		f1dom2g 7526	. . . . 5 |- ( ( A e. _V /\ ran f e. _V /\ f : A -1-1-> ran f ) -> A ~<_ ran f )
15	6, 9, 13, 14	syl3anc 1226	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A ~<_ ran f )
16		sdomnen 7537	. . . . . . . 8 |- ( A ~< B -> -. A ~~ B )
17	16	adantr 463	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> -. A ~~ B )
18		ssdif0 3873	. . . . . . . 8 |- ( B C_ ran f <-> ( B \ ran f ) = (/) )
19		simplr 753	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-> B )
20		f1f 5763	. . . . . . . . . . . . . . 15 |- ( f : A -1-1-> B -> f : A --> B )
21		df-f 5574	. . . . . . . . . . . . . . 15 |- ( f : A --> B <-> ( f Fn A /\ ran f C_ B ) )
22	20, 21	sylib 196	. . . . . . . . . . . . . 14 |- ( f : A -1-1-> B -> ( f Fn A /\ ran f C_ B ) )
23	22	simprd 461	. . . . . . . . . . . . 13 |- ( f : A -1-1-> B -> ran f C_ B )
24	19, 23	syl 16	. . . . . . . . . . . 12 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f C_ B )
25		simpr 459	. . . . . . . . . . . 12 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> B C_ ran f )
26	24, 25	eqssd 3506	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f = B )
27		dff1o5 5807	. . . . . . . . . . 11 |- ( f : A -1-1-onto-> B <-> ( f : A -1-1-> B /\ ran f = B ) )
28	19, 26, 27	sylanbrc 662	. . . . . . . . . 10 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-onto-> B )
29		f1oen3g 7524	. . . . . . . . . 10 |- ( ( f e. _V /\ f : A -1-1-onto-> B ) -> A ~~ B )
30	7, 28, 29	sylancr 661	. . . . . . . . 9 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> A ~~ B )
31	30	ex 432	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( B C_ ran f -> A ~~ B ) )
32	18, 31	syl5bir 218	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ( B \ ran f ) = (/) -> A ~~ B ) )
33	17, 32	mtod 177	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> -. ( B \ ran f ) = (/) )
34		neq0 3794	. . . . . 6 |- ( -. ( B \ ran f ) = (/) <-> E. w w e. ( B \ ran f ) )
35	33, 34	sylib 196	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> E. w w e. ( B \ ran f ) )
36		snssi 4160	. . . . . . 7 |- ( w e. ( B \ ran f ) -> { w } C_ ( B \ ran f ) )
37		vex 3109	. . . . . . . . 9 |- w e. _V
38		en2sn 7588	. . . . . . . . 9 |- ( ( A e. _V /\ w e. _V ) -> { A } ~~ { w } )
39	6, 37, 38	sylancl 660	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> { A } ~~ { w } )
40	4	brrelex2i 5030	. . . . . . . . . 10 |- ( A ~< B -> B e. _V )
41	40	adantr 463	. . . . . . . . 9 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B e. _V )
42		difexg 4585	. . . . . . . . 9 |- ( B e. _V -> ( B \ ran f ) e. _V )
43		ssdomg 7554	. . . . . . . . 9 |- ( ( B \ ran f ) e. _V -> ( { w } C_ ( B \ ran f ) -> { w } ~<_ ( B \ ran f ) ) )
44	41, 42, 43	3syl 20	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( { w } C_ ( B \ ran f ) -> { w } ~<_ ( B \ ran f ) ) )
45		endomtr 7566	. . . . . . . 8 |- ( ( { A } ~~ { w } /\ { w } ~<_ ( B \ ran f ) ) -> { A } ~<_ ( B \ ran f ) )
46	39, 44, 45	syl6an 543	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( { w } C_ ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) )
47	36, 46	syl5 32	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( w e. ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) )
48	47	exlimdv 1729	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( E. w w e. ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) )
49	35, 48	mpd 15	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> { A } ~<_ ( B \ ran f ) )
50		disjdif 3888	. . . . 5 |- ( ran f i^i ( B \ ran f ) ) = (/)
51	50	a1i 11	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ran f i^i ( B \ ran f ) ) = (/) )
52		undom 7598	. . . 4 |- ( ( ( A ~<_ ran f /\ { A } ~<_ ( B \ ran f ) ) /\ ( ran f i^i ( B \ ran f ) ) = (/) ) -> ( A u. { A } ) ~<_ ( ran f u. ( B \ ran f ) ) )
53	15, 49, 51, 52	syl21anc 1225	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( A u. { A } ) ~<_ ( ran f u. ( B \ ran f ) ) )
54		df-suc 4873	. . . 4 |- suc A = ( A u. { A } )
55	54	a1i 11	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> suc A = ( A u. { A } ) )
56		undif2 3892	. . . 4 |- ( ran f u. ( B \ ran f ) ) = ( ran f u. B )
57	23	adantl 464	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f C_ B )
58		ssequn1 3660	. . . . 5 |- ( ran f C_ B <-> ( ran f u. B ) = B )
59	57, 58	sylib 196	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ran f u. B ) = B )
60	56, 59	syl5req 2508	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B = ( ran f u. ( B \ ran f ) ) )
61	53, 55, 60	3brtr4d 4469	. 2 |- ( ( A ~< B /\ f : A -1-1-> B ) -> suc A ~<_ B )
62	3, 61	exlimddv 1731	1 |- ( A ~< B -> suc A ~<_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   = wceq 1398   E.wex 1617   e. wcel 1823   _Vcvv 3106   \ cdif 3458   u. cun 3459   i^i cin 3460   C_ wss 3461   (/)c0 3783   {csn 4016   class class class wbr 4439   suc csuc 4869   ran crn 4989   Fn wfn 5565   -->wf 5566   -1-1->wf1 5567   -1-1-onto->wf1o 5569   ~~ cen 7506   ~<_ cdom 7507   ~< csdm 7508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512

This theorem is referenced by:  sucdom  7708  card2inf  7973