Theorem ssdomg 7055

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 4267	. . 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 5674	. . . . . . . . . 10 |- ( _I |` A ) : A -1-1-onto-> A
4		dff1o3 5640	. . . . . . . . . 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 5610	. . . . . . . 8 |- ( ( _I |` A ) : A -onto-> A -> ( _I |` A ) : A --> A )
8	6, 7	ax-mp 5	. . . . . . 7 |- ( _I |` A ) : A --> A
9		fss 5541	. . . . . . 7 |- ( ( ( _I |` A ) : A --> A /\ A C_ B ) -> ( _I |` A ) : A --> B )
10	8, 9	mpan 428	. . . . . 6 |- ( A C_ B -> ( _I |` A ) : A --> B )
11		funi 5404	. . . . . . . 8 |- Fun _I
12		cnvi 5187	. . . . . . . . 9 |- `' _I = _I
13	12	funeqi 5393	. . . . . . . 8 |- ( Fun `' _I <-> Fun _I )
14	11, 13	mpbir 146	. . . . . . 7 |- Fun `' _I
15		funres11 5448	. . . . . . 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 5377	. . . . 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 7032	. . 3 |- ( ( A e. _V /\ B e. V /\ ( _I |` A ) : A -1-1-> B ) -> A ~<_ B )
22	1, 2, 20, 21	syl3anc 1278	. 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 2209   _Vcvv 2821   C_ wss 3220   class class class wbr 4125   _I cid 4428   `'ccnv 4768   |` cres 4771   Fun wfun 5366   -->wf 5368   -1-1->wf1 5369   -onto->wfo 5370   -1-1-onto->wf1o 5371   ~<_ cdom 7011

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-dom 7014

This theorem is referenced by:  cnvct  7087  ssct  7104  xpdom3m  7122  0domg  7127  mapdom1g  7137  phplem4dom  7153  nndomo  7155  phpm  7157  fict  7160  domfiexmid  7172  infnfi  7189  exmidfodomrlemr  7544  exmidfodomrlemrALT  7545  pw1dom2  7576  fihashss  11235  phicl2  12970  phibnd  12973  4sqlem11  13158  qnnen  13300  isnzr2  14464  sbthom  16976