Theorem sucdom2 7768

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 7597	. . 3 |- ( A ~< B -> A ~<_ B )
2		brdomi 7580	. . 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 7576	. . . . . . 7 |- Rel ~<
5	4	brrelexi 4875	. . . . . 6 |- ( A ~< B -> A e. _V )
6	5	adantr 467	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A e. _V )
7		vex 3048	. . . . . . 7 |- f e. _V
8	7	rnex 6727	. . . . . 6 |- ran f e. _V
9	8	a1i 11	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f e. _V )
10		f1f1orn 5825	. . . . . . 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 5813	. . . . . 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 7587	. . . . 5 |- ( ( A e. _V /\ ran f e. _V /\ f : A -1-1-> ran f ) -> A ~<_ ran f )
15	6, 9, 13, 14	syl3anc 1268	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A ~<_ ran f )
16		sdomnen 7598	. . . . . . . 8 |- ( A ~< B -> -. A ~~ B )
17	16	adantr 467	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> -. A ~~ B )
18		ssdif0 3823	. . . . . . . 8 |- ( B C_ ran f <-> ( B \ ran f ) = (/) )
19		simplr 762	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-> B )
20		f1f 5779	. . . . . . . . . . . . . . 15 |- ( f : A -1-1-> B -> f : A --> B )
21		df-f 5586	. . . . . . . . . . . . . . 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 3449	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f = B )
27		dff1o5 5823	. . . . . . . . . . 11 |- ( f : A -1-1-onto-> B <-> ( f : A -1-1-> B /\ ran f = B ) )
28	19, 26, 27	sylanbrc 670	. . . . . . . . . 10 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-onto-> B )
29		f1oen3g 7585	. . . . . . . . . 10 |- ( ( f e. _V /\ f : A -1-1-onto-> B ) -> A ~~ B )
30	7, 28, 29	sylancr 669	. . . . . . . . 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 3742	. . . . . 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 4116	. . . . . . 7 |- ( w e. ( B \ ran f ) -> { w } C_ ( B \ ran f ) )
37		vex 3048	. . . . . . . . 9 |- w e. _V
38		en2sn 7649	. . . . . . . . 9 |- ( ( A e. _V /\ w e. _V ) -> { A } ~~ { w } )
39	6, 37, 38	sylancl 668	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> { A } ~~ { w } )
40	4	brrelex2i 4876	. . . . . . . . . 10 |- ( A ~< B -> B e. _V )
41	40	adantr 467	. . . . . . . . 9 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B e. _V )
42		difexg 4551	. . . . . . . . 9 |- ( B e. _V -> ( B \ ran f ) e. _V )
43		ssdomg 7615	. . . . . . . . 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 7627	. . . . . . . 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 33	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( w e. ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) )
48	47	exlimdv 1779	. . . . 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 3839	. . . . 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 7660	. . . 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 1267	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( A u. { A } ) ~<_ ( ran f u. ( B \ ran f ) ) )
54		df-suc 5429	. . . 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 3843	. . . 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 3604	. . . . 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 2498	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B = ( ran f u. ( B \ ran f ) ) )
61	53, 55, 60	3brtr4d 4433	. 2 |- ( ( A ~< B /\ f : A -1-1-> B ) -> suc A ~<_ B )
62	3, 61	exlimddv 1781	1 |- ( A ~< B -> suc A ~<_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   _Vcvv 3045   \ cdif 3401   u. cun 3402   i^i cin 3403   C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   ran crn 4835   suc csuc 5425   Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -1-1-onto->wf1o 5581   ~~ cen 7566   ~<_ cdom 7567   ~< csdm 7568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-suc 5429  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572

This theorem is referenced by:  sucdom  7769  card2inf  8070