Theorem phplem4 7012

Description: Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)

Hypotheses

Ref	Expression
phplem2.1	|- A e. _V
phplem2.2	|- B e. _V

Assertion

Ref	Expression
phplem4	|- ( ( A e. om /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )

Proof of Theorem phplem4

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6893	. 2 |- ( suc A ~~ suc B <-> E. f f : suc A -1-1-onto-> suc B )
2		f1of1 5570	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -1-1-> suc B )
3	2	adantl 277	. . . . . . . . 9 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> f : suc A -1-1-> suc B )
4		phplem2.2	. . . . . . . . . 10 |- B e. _V
5	4	sucex 4590	. . . . . . . . 9 |- suc B e. _V
6		sssucid 4505	. . . . . . . . . 10 |- A C_ suc A
7		phplem2.1	. . . . . . . . . 10 |- A e. _V
8		f1imaen2g 6943	. . . . . . . . . 10 |- ( ( ( f : suc A -1-1-> suc B /\ suc B e. _V ) /\ ( A C_ suc A /\ A e. _V ) ) -> ( f " A ) ~~ A )
9	6, 7, 8	mpanr12 439	. . . . . . . . 9 |- ( ( f : suc A -1-1-> suc B /\ suc B e. _V ) -> ( f " A ) ~~ A )
10	3, 5, 9	sylancl 413	. . . . . . . 8 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) ~~ A )
11	10	ensymd 6933	. . . . . . 7 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( f " A ) )
12		nnord 4703	. . . . . . . . . 10 |- ( A e. om -> Ord A )
13		orddif 4638	. . . . . . . . . 10 |- ( Ord A -> A = ( suc A \ { A } ) )
14	12, 13	syl 14	. . . . . . . . 9 |- ( A e. om -> A = ( suc A \ { A } ) )
15	14	imaeq2d 5067	. . . . . . . 8 |- ( A e. om -> ( f " A ) = ( f " ( suc A \ { A } ) ) )
16		f1ofn 5572	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> f Fn suc A )
17	7	sucid 4507	. . . . . . . . . . 11 |- A e. suc A
18		fnsnfv 5692	. . . . . . . . . . 11 |- ( ( f Fn suc A /\ A e. suc A ) -> { ( f ` A ) } = ( f " { A } ) )
19	16, 17, 18	sylancl 413	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> { ( f ` A ) } = ( f " { A } ) )
20	19	difeq2d 3322	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( ( f " suc A ) \ { ( f ` A ) } ) = ( ( f " suc A ) \ ( f " { A } ) ) )
21		imadmrn 5077	. . . . . . . . . . . 12 |- ( f " dom f ) = ran f
22	21	eqcomi 2233	. . . . . . . . . . 11 |- ran f = ( f " dom f )
23		f1ofo 5578	. . . . . . . . . . . 12 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -onto-> suc B )
24		forn 5550	. . . . . . . . . . . 12 |- ( f : suc A -onto-> suc B -> ran f = suc B )
25	23, 24	syl 14	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> ran f = suc B )
26		f1odm 5575	. . . . . . . . . . . 12 |- ( f : suc A -1-1-onto-> suc B -> dom f = suc A )
27	26	imaeq2d 5067	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> ( f " dom f ) = ( f " suc A ) )
28	22, 25, 27	3eqtr3a 2286	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> suc B = ( f " suc A ) )
29	28	difeq1d 3321	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( suc B \ { ( f ` A ) } ) = ( ( f " suc A ) \ { ( f ` A ) } ) )
30		dff1o3 5577	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B <-> ( f : suc A -onto-> suc B /\ Fun `' f ) )
31	30	simprbi 275	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> Fun `' f )
32		imadif 5400	. . . . . . . . . 10 |- ( Fun `' f -> ( f " ( suc A \ { A } ) ) = ( ( f " suc A ) \ ( f " { A } ) ) )
33	31, 32	syl 14	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( f " ( suc A \ { A } ) ) = ( ( f " suc A ) \ ( f " { A } ) ) )
34	20, 29, 33	3eqtr4rd 2273	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( f " ( suc A \ { A } ) ) = ( suc B \ { ( f ` A ) } ) )
35	15, 34	sylan9eq 2282	. . . . . . 7 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) = ( suc B \ { ( f ` A ) } ) )
36	11, 35	breqtrd 4108	. . . . . 6 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( suc B \ { ( f ` A ) } ) )
37		fnfvelrn 5766	. . . . . . . . . 10 |- ( ( f Fn suc A /\ A e. suc A ) -> ( f ` A ) e. ran f )
38	16, 17, 37	sylancl 413	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( f ` A ) e. ran f )
39	24	eleq2d 2299	. . . . . . . . . 10 |- ( f : suc A -onto-> suc B -> ( ( f ` A ) e. ran f <-> ( f ` A ) e. suc B ) )
40	23, 39	syl 14	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( ( f ` A ) e. ran f <-> ( f ` A ) e. suc B ) )
41	38, 40	mpbid 147	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( f ` A ) e. suc B )
42		vex 2802	. . . . . . . . . 10 |- f e. _V
43	42, 7	fvex 5646	. . . . . . . . 9 |- ( f ` A ) e. _V
44	4, 43	phplem3 7011	. . . . . . . 8 |- ( ( B e. om /\ ( f ` A ) e. suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
45	41, 44	sylan2 286	. . . . . . 7 |- ( ( B e. om /\ f : suc A -1-1-onto-> suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
46	45	ensymd 6933	. . . . . 6 |- ( ( B e. om /\ f : suc A -1-1-onto-> suc B ) -> ( suc B \ { ( f ` A ) } ) ~~ B )
47		entr 6934	. . . . . 6 |- ( ( A ~~ ( suc B \ { ( f ` A ) } ) /\ ( suc B \ { ( f ` A ) } ) ~~ B ) -> A ~~ B )
48	36, 46, 47	syl2an 289	. . . . 5 |- ( ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) /\ ( B e. om /\ f : suc A -1-1-onto-> suc B ) ) -> A ~~ B )
49	48	anandirs 595	. . . 4 |- ( ( ( A e. om /\ B e. om ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ B )
50	49	ex 115	. . 3 |- ( ( A e. om /\ B e. om ) -> ( f : suc A -1-1-onto-> suc B -> A ~~ B ) )
51	50	exlimdv 1865	. 2 |- ( ( A e. om /\ B e. om ) -> ( E. f f : suc A -1-1-onto-> suc B -> A ~~ B ) )
52	1, 51	biimtrid 152	1 |- ( ( A e. om /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   \ cdif 3194   C_ wss 3197   {csn 3666   class class class wbr 4082   Ord word 4452   suc csuc 4455   omcom 4681   `'ccnv 4717   dom cdm 4718   ran crn 4719   "cima 4721   Fun wfun 5311   Fn wfn 5312   -1-1->wf1 5314   -onto->wfo 5315   -1-1-onto->wf1o 5316   ` cfv 5317   ~~ cen 6883

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-in1 617  ax-in2 618  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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  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-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-er 6678  df-en 6886

This theorem is referenced by:  nneneq  7014  php5  7015