Theorem sucdom2 7772

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 7602	. . 3 |- ( A ~< B -> A ~<_ B )
2		brdomi 7586	. . 3 |- ( A ~<_ B -> E. f f : A -1-1-> B )
3	1, 2	syl 17	. 2 |- ( A ~< B -> E. f f : A -1-1-> B )
4		relsdom 7582	. . . . . . 7 |- Rel ~<
5	4	brrelexi 4892	. . . . . 6 |- ( A ~< B -> A e. _V )
6	5	adantr 467	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A e. _V )
7		vex 3085	. . . . . . 7 |- f e. _V
8	7	rnex 6739	. . . . . 6 |- ran f e. _V
9	8	a1i 11	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f e. _V )
10		f1f1orn 5840	. . . . . . 7 |- ( f : A -1-1-> B -> f : A -1-1-onto-> ran f )
11	10	adantl 468	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> f : A -1-1-onto-> ran f )
12		f1of1 5828	. . . . . 6 |- ( f : A -1-1-onto-> ran f -> f : A -1-1-> ran f )
13	11, 12	syl 17	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> f : A -1-1-> ran f )
14		f1dom2g 7592	. . . . 5 |- ( ( A e. _V /\ ran f e. _V /\ f : A -1-1-> ran f ) -> A ~<_ ran f )
15	6, 9, 13, 14	syl3anc 1265	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A ~<_ ran f )
16		sdomnen 7603	. . . . . . . 8 |- ( A ~< B -> -. A ~~ B )
17	16	adantr 467	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> -. A ~~ B )
18		ssdif0 3852	. . . . . . . 8 |- ( B C_ ran f <-> ( B \ ran f ) = (/) )
19		simplr 761	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-> B )
20		f1f 5794	. . . . . . . . . . . . . . 15 |- ( f : A -1-1-> B -> f : A --> B )
21		df-f 5603	. . . . . . . . . . . . . . 15 |- ( f : A --> B <-> ( f Fn A /\ ran f C_ B ) )
22	20, 21	sylib 200	. . . . . . . . . . . . . 14 |- ( f : A -1-1-> B -> ( f Fn A /\ ran f C_ B ) )
23	22	simprd 465	. . . . . . . . . . . . 13 |- ( f : A -1-1-> B -> ran f C_ B )
24	19, 23	syl 17	. . . . . . . . . . . 12 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f C_ B )
25		simpr 463	. . . . . . . . . . . 12 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> B C_ ran f )
26	24, 25	eqssd 3482	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f = B )
27		dff1o5 5838	. . . . . . . . . . 11 |- ( f : A -1-1-onto-> B <-> ( f : A -1-1-> B /\ ran f = B ) )
28	19, 26, 27	sylanbrc 669	. . . . . . . . . 10 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-onto-> B )
29		f1oen3g 7590	. . . . . . . . . 10 |- ( ( f e. _V /\ f : A -1-1-onto-> B ) -> A ~~ B )
30	7, 28, 29	sylancr 668	. . . . . . . . 9 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> A ~~ B )
31	30	ex 436	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( B C_ ran f -> A ~~ B ) )
32	18, 31	syl5bir 222	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ( B \ ran f ) = (/) -> A ~~ B ) )
33	17, 32	mtod 181	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> -. ( B \ ran f ) = (/) )
34		neq0 3773	. . . . . 6 |- ( -. ( B \ ran f ) = (/) <-> E. w w e. ( B \ ran f ) )
35	33, 34	sylib 200	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> E. w w e. ( B \ ran f ) )
36		snssi 4142	. . . . . . 7 |- ( w e. ( B \ ran f ) -> { w } C_ ( B \ ran f ) )
37		vex 3085	. . . . . . . . 9 |- w e. _V
38		en2sn 7654	. . . . . . . . 9 |- ( ( A e. _V /\ w e. _V ) -> { A } ~~ { w } )
39	6, 37, 38	sylancl 667	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> { A } ~~ { w } )
40	4	brrelex2i 4893	. . . . . . . . . 10 |- ( A ~< B -> B e. _V )
41	40	adantr 467	. . . . . . . . 9 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B e. _V )
42		difexg 4570	. . . . . . . . 9 |- ( B e. _V -> ( B \ ran f ) e. _V )
43		ssdomg 7620	. . . . . . . . 9 |- ( ( B \ ran f ) e. _V -> ( { w } C_ ( B \ ran f ) -> { w } ~<_ ( B \ ran f ) ) )
44	41, 42, 43	3syl 18	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( { w } C_ ( B \ ran f ) -> { w } ~<_ ( B \ ran f ) ) )
45		endomtr 7632	. . . . . . . 8 |- ( ( { A } ~~ { w } /\ { w } ~<_ ( B \ ran f ) ) -> { A } ~<_ ( B \ ran f ) )
46	39, 44, 45	syl6an 548	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( { w } C_ ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) )
47	36, 46	syl5 34	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( w e. ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) )
48	47	exlimdv 1769	. . . . 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 3868	. . . . 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 7664	. . . 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 1264	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( A u. { A } ) ~<_ ( ran f u. ( B \ ran f ) ) )
54		df-suc 5446	. . . 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 3872	. . . 4 |- ( ran f u. ( B \ ran f ) ) = ( ran f u. B )
57	23	adantl 468	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f C_ B )
58		ssequn1 3637	. . . . 5 |- ( ran f C_ B <-> ( ran f u. B ) = B )
59	57, 58	sylib 200	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ran f u. B ) = B )
60	56, 59	syl5req 2477	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B = ( ran f u. ( B \ ran f ) ) )
61	53, 55, 60	3brtr4d 4452	. 2 |- ( ( A ~< B /\ f : A -1-1-> B ) -> suc A ~<_ B )
62	3, 61	exlimddv 1771	1 |- ( A ~< B -> suc A ~<_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1438   E.wex 1660   e. wcel 1869   _Vcvv 3082   \ cdif 3434   u. cun 3435   i^i cin 3436   C_ wss 3437   (/)c0 3762   {csn 3997   class class class wbr 4421   ran crn 4852   suc csuc 5442   Fn wfn 5594   -->wf 5595   -1-1->wf1 5596   -1-1-onto->wf1o 5598   ~~ cen 7572   ~<_ cdom 7573   ~< csdm 7574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-suc 5446  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-1o 7188  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578

This theorem is referenced by:  sucdom  7773  card2inf  8074