Theorem sucdom2 7503

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 7333	. . 3 |- ( A ~< B -> A ~<_ B )
2		brdomi 7317	. . 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 7313	. . . . . . 7 |- Rel ~<
5	4	brrelexi 4875	. . . . . 6 |- ( A ~< B -> A e. _V )
6	5	adantr 462	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A e. _V )
7		vex 2973	. . . . . . 7 |- f e. _V
8	7	rnex 6511	. . . . . 6 |- ran f e. _V
9	8	a1i 11	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f e. _V )
10		f1f1orn 5649	. . . . . . 7 |- ( f : A -1-1-> B -> f : A -1-1-onto-> ran f )
11	10	adantl 463	. . . . . 6 |- ( ( A ~< B /\ f : A -1-1-> B ) -> f : A -1-1-onto-> ran f )
12		f1of1 5637	. . . . . 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 7323	. . . . 5 |- ( ( A e. _V /\ ran f e. _V /\ f : A -1-1-> ran f ) -> A ~<_ ran f )
15	6, 9, 13, 14	syl3anc 1213	. . . 4 |- ( ( A ~< B /\ f : A -1-1-> B ) -> A ~<_ ran f )
16		sdomnen 7334	. . . . . . . 8 |- ( A ~< B -> -. A ~~ B )
17	16	adantr 462	. . . . . . 7 |- ( ( A ~< B /\ f : A -1-1-> B ) -> -. A ~~ B )
18		ssdif0 3734	. . . . . . . 8 |- ( B C_ ran f <-> ( B \ ran f ) = (/) )
19		simplr 749	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-> B )
20		f1f 5603	. . . . . . . . . . . . . . 15 |- ( f : A -1-1-> B -> f : A --> B )
21		df-f 5419	. . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . 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 458	. . . . . . . . . . . 12 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> B C_ ran f )
26	24, 25	eqssd 3370	. . . . . . . . . . 11 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f = B )
27		dff1o5 5647	. . . . . . . . . . 11 |- ( f : A -1-1-onto-> B <-> ( f : A -1-1-> B /\ ran f = B ) )
28	19, 26, 27	sylanbrc 659	. . . . . . . . . 10 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-onto-> B )
29		f1oen3g 7321	. . . . . . . . . 10 |- ( ( f e. _V /\ f : A -1-1-onto-> B ) -> A ~~ B )
30	7, 28, 29	sylancr 658	. . . . . . . . 9 |- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> A ~~ B )
31	30	ex 434	. . . . . . . 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 3644	. . . . . 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 4014	. . . . . . 7 |- ( w e. ( B \ ran f ) -> { w } C_ ( B \ ran f ) )
37		vex 2973	. . . . . . . . 9 |- w e. _V
38		en2sn 7385	. . . . . . . . 9 |- ( ( A e. _V /\ w e. _V ) -> { A } ~~ { w } )
39	6, 37, 38	sylancl 657	. . . . . . . 8 |- ( ( A ~< B /\ f : A -1-1-> B ) -> { A } ~~ { w } )
40	4	brrelex2i 4876	. . . . . . . . . 10 |- ( A ~< B -> B e. _V )
41	40	adantr 462	. . . . . . . . 9 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B e. _V )
42		difexg 4437	. . . . . . . . 9 |- ( B e. _V -> ( B \ ran f ) e. _V )
43		ssdomg 7351	. . . . . . . . 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 7363	. . . . . . . 8 |- ( ( { A } ~~ { w } /\ { w } ~<_ ( B \ ran f ) ) -> { A } ~<_ ( B \ ran f ) )
46	39, 44, 45	syl6an 542	. . . . . . 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 1695	. . . . 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 3748	. . . . 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 7395	. . . 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 1212	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ( A u. { A } ) ~<_ ( ran f u. ( B \ ran f ) ) )
54		df-suc 4721	. . . 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 3752	. . . 4 |- ( ran f u. ( B \ ran f ) ) = ( ran f u. B )
57	23	adantl 463	. . . . 5 |- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f C_ B )
58		ssequn1 3523	. . . . 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 2486	. . 3 |- ( ( A ~< B /\ f : A -1-1-> B ) -> B = ( ran f u. ( B \ ran f ) ) )
61	53, 55, 60	3brtr4d 4319	. 2 |- ( ( A ~< B /\ f : A -1-1-> B ) -> suc A ~<_ B )
62	3, 61	exlimddv 1697	1 |- ( A ~< B -> suc A ~<_ B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1364   E.wex 1591   e. wcel 1761   _Vcvv 2970   \ cdif 3322   u. cun 3323   i^i cin 3324   C_ wss 3325   (/)c0 3634   {csn 3874   class class class wbr 4289   suc csuc 4717   ran crn 4837   Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -1-1-onto->wf1o 5414   ~~ cen 7303   ~<_ cdom 7304   ~< csdm 7305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309

This theorem is referenced by:  sucdom  7504  card2inf  7766