Theorem sucdom2 7262

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 7094	. . 3 |- ( A ~< B -> A ~<_ B )
2		brdomi 7078	. . 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 7075	. . . . . . 7 |- Rel ~<
5	4	brrelexi 4877	. . . . . 6 |- ( A ~< B -> A e. _V )
6	5	adantr 452	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A e. _V )
7		vex 2919	. . . . . . 7 |- f e. _V
8	7	rnex 5092	. . . . . 6 |- ran f e. _V
9	8	a1i 11	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f e. _V )
10		f1f1orn 5644	. . . . . . 7 |- ( f : A -1-1-> B -> f : A -1-1-onto-> ran f )
11	10	adantl 453	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> f : A -1-1-onto-> ran f )
12		f1of1 5632	. . . . . 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 7084	. . . . 5 |- ( ( A e. _V /\ ran f e. _V /\ f : A -1-1-> ran f ) -> A ~<_ ran f )
15	6, 9, 13, 14	syl3anc 1184	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A ~<_ ran f )
16		sdomnen 7095	. . . . . . . 8 |- ( A ~< B -> -. A ~~ B )
17	16	adantr 452	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> -. A ~~ B )
18		ssdif0 3646	. . . . . . . 8 |- ( B C_ ran f <-> ( B \ ran f ) = (/) )
19		simplr 732	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-> B )
20		f1f 5598	. . . . . . . . . . . . . . 15 |- ( f : A -1-1-> B -> f : A --> B )
21		df-f 5417	. . . . . . . . . . . . . . 15 |- ( f : A --> B <-> ( f Fn A /\ ran f C_ B ) )
22	20, 21	sylib 189	. . . . . . . . . . . . . 14 |- ( f : A -1-1-> B -> ( f Fn A /\ ran f C_ B ) )
23	22	simprd 450	. . . . . . . . . . . . 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 448	. . . . . . . . . . . 12 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> B C_ ran f )
26	24, 25	eqssd 3325	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f = B )
27		dff1o5 5642	. . . . . . . . . . 11 |- ( f : A -1-1-onto-> B <-> ( f : A -1-1-> B /\ ran f = B ) )
28	19, 26, 27	sylanbrc 646	. . . . . . . . . 10 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-onto-> B )
29		f1oen3g 7082	. . . . . . . . . 10 |- ( ( f e. _V /\ f : A -1-1-onto-> B ) -> A ~~ B )
30	7, 28, 29	sylancr 645	. . . . . . . . 9 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> A ~~ B )
31	30	ex 424	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( B C_ ran f -> A ~~ B ) )
32	18, 31	syl5bir 210	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ( B \ ran f ) = (/) -> A ~~ B ) )
33	17, 32	mtod 170	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> -. ( B \ ran f ) = (/) )
34		neq0 3598	. . . . . 6 |- ( -. ( B \ ran f ) = (/) <-> E. w w e. ( B \ ran f ) )
35	33, 34	sylib 189	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> E. w w e. ( B \ ran f ) )
36		snssi 3902	. . . . . . 7 |- ( w e. ( B \ ran f ) -> { w } C_ ( B \ ran f ) )
37		vex 2919	. . . . . . . . 9 |- w e. _V
38		en2sn 7145	. . . . . . . . 9 |- ( ( A e. _V /\ w e. _V ) -> { A } ~~ { w } )
39	6, 37, 38	sylancl 644	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> { A } ~~ { w } )
40	4	brrelex2i 4878	. . . . . . . . . 10 |- ( A ~< B -> B e. _V )
41	40	adantr 452	. . . . . . . . 9 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B e. _V )
42		difexg 4311	. . . . . . . . 9 |- ( B e. _V -> ( B \ ran f ) e. _V )
43		ssdomg 7112	. . . . . . . . 9 |- ( ( B \ ran f ) e. _V -> ( { w } C_ ( B \ ran f ) -> { w } ~<_ ( B \ ran f ) ) )
44	41, 42, 43	3syl 19	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( { w } C_ ( B \ ran f ) -> { w } ~<_ ( B \ ran f ) ) )
45		endomtr 7124	. . . . . . . 8 |- ( ( { A } ~~ { w } /\ { w } ~<_ ( B \ ran f ) ) -> { A } ~<_ ( B \ ran f ) )
46	39, 44, 45	ee12an 1369	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( { w } C_ ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) )
47	36, 46	syl5 30	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( w e. ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) )
48	47	exlimdv 1643	. . . . 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 3660	. . . . 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 7155	. . . 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 1183	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( A u. { A } ) ~<_ ( ran f u. ( B \ ran f ) ) )
54		df-suc 4547	. . . 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 3664	. . . 4 |- ( ran f u. ( B \ ran f ) ) = ( ran f u. B )
57	23	adantl 453	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f C_ B )
58		ssequn1 3477	. . . . 5 |- ( ran f C_ B <-> ( ran f u. B ) = B )
59	57, 58	sylib 189	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ran f u. B ) = B )
60	56, 59	syl5req 2449	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B = ( ran f u. ( B \ ran f ) ) )
61	53, 55, 60	3brtr4d 4202	. 2 |- ( ( A ~< B /\ f : A -1-1-> B ) -> suc A ~<_ B )
62	3, 61	exlimddv 1645	1 |- ( A ~< B -> suc A ~<_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   E.wex 1547   = wceq 1649   e. wcel 1721   _Vcvv 2916   \ cdif 3277   u. cun 3278   i^i cin 3279   C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172   suc csuc 4543   ran crn 4838   Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ~~ cen 7065   ~<_ cdom 7066   ~< csdm 7067

This theorem is referenced by:  sucdom  7263  card2inf  7479

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071