Theorem ssdomg 6930

Description: A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
ssdomg	|- ( B e. V -> ( A C_ B -> A ~<_ B ) )

Proof of Theorem ssdomg

Step	Hyp	Ref	Expression
1		ssexg 4223	. . 3 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
2		simpr 110	. . 3 |- ( ( A C_ B /\ B e. V ) -> B e. V )
3		f1oi 5611	. . . . . . . . . 10 |- ( _I |` A ) : A -1-1-onto-> A
4		dff1o3 5578	. . . . . . . . . 10 |- ( ( _I |` A ) : A -1-1-onto-> A <-> ( ( _I |` A ) : A -onto-> A /\ Fun `' ( _I |` A ) ) )
5	3, 4	mpbi 145	. . . . . . . . 9 |- ( ( _I |` A ) : A -onto-> A /\ Fun `' ( _I |` A ) )
6	5	simpli 111	. . . . . . . 8 |- ( _I |` A ) : A -onto-> A
7		fof 5548	. . . . . . . 8 |- ( ( _I |` A ) : A -onto-> A -> ( _I |` A ) : A --> A )
8	6, 7	ax-mp 5	. . . . . . 7 |- ( _I |` A ) : A --> A
9		fss 5485	. . . . . . 7 |- ( ( ( _I |` A ) : A --> A /\ A C_ B ) -> ( _I |` A ) : A --> B )
10	8, 9	mpan 424	. . . . . 6 |- ( A C_ B -> ( _I |` A ) : A --> B )
11		funi 5350	. . . . . . . 8 |- Fun _I
12		cnvi 5133	. . . . . . . . 9 |- `' _I = _I
13	12	funeqi 5339	. . . . . . . 8 |- ( Fun `' _I <-> Fun _I )
14	11, 13	mpbir 146	. . . . . . 7 |- Fun `' _I
15		funres11 5393	. . . . . . 7 |- ( Fun `' _I -> Fun `' ( _I |` A ) )
16	14, 15	ax-mp 5	. . . . . 6 |- Fun `' ( _I |` A )
17	10, 16	jctir 313	. . . . 5 |- ( A C_ B -> ( ( _I |` A ) : A --> B /\ Fun `' ( _I |` A ) ) )
18		df-f1 5323	. . . . 5 |- ( ( _I |` A ) : A -1-1-> B <-> ( ( _I |` A ) : A --> B /\ Fun `' ( _I |` A ) ) )
19	17, 18	sylibr 134	. . . 4 |- ( A C_ B -> ( _I |` A ) : A -1-1-> B )
20	19	adantr 276	. . 3 |- ( ( A C_ B /\ B e. V ) -> ( _I |` A ) : A -1-1-> B )
21		f1dom2g 6907	. . 3 |- ( ( A e. _V /\ B e. V /\ ( _I |` A ) : A -1-1-> B ) -> A ~<_ B )
22	1, 2, 20, 21	syl3anc 1271	. 2 |- ( ( A C_ B /\ B e. V ) -> A ~<_ B )
23	22	expcom 116	1 |- ( B e. V -> ( A C_ B -> A ~<_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   _Vcvv 2799   C_ wss 3197   class class class wbr 4083   _I cid 4379   `'ccnv 4718   |` cres 4721   Fun wfun 5312   -->wf 5314   -1-1->wf1 5315   -onto->wfo 5316   -1-1-onto->wf1o 5317   ~<_ cdom 6886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-dom 6889

This theorem is referenced by:  cnvct  6962  ssct  6975  xpdom3m  6993  0domg  6998  mapdom1g  7008  phplem4dom  7023  nndomo  7025  phpm  7027  fict  7030  domfiexmid  7040  infnfi  7057  exmidfodomrlemr  7380  exmidfodomrlemrALT  7381  pw1dom2  7412  fihashss  11038  phicl2  12736  phibnd  12739  4sqlem11  12924  qnnen  13002  isnzr2  14148  sbthom  16394