Theorem phplem4 8142

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 7964	. 2 |- ( suc A ~~ suc B <-> E. f f : suc A -1-1-onto-> suc B )
2		f1of1 6136	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -1-1-> suc B )
3	2	adantl 482	. . . . . . . . 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 7011	. . . . . . . . 9 |- suc B e. _V
6		sssucid 5802	. . . . . . . . . 10 |- A C_ suc A
7		phplem2.1	. . . . . . . . . 10 |- A e. _V
8		f1imaen2g 8017	. . . . . . . . . 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 721	. . . . . . . . 9 |- ( ( f : suc A -1-1-> suc B /\ suc B e. _V ) -> ( f " A ) ~~ A )
10	3, 5, 9	sylancl 694	. . . . . . . 8 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) ~~ A )
11	10	ensymd 8007	. . . . . . 7 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( f " A ) )
12		nnord 7073	. . . . . . . . . 10 |- ( A e. om -> Ord A )
13		orddif 5820	. . . . . . . . . 10 |- ( Ord A -> A = ( suc A \ { A } ) )
14	12, 13	syl 17	. . . . . . . . 9 |- ( A e. om -> A = ( suc A \ { A } ) )
15	14	imaeq2d 5466	. . . . . . . 8 |- ( A e. om -> ( f " A ) = ( f " ( suc A \ { A } ) ) )
16		f1ofn 6138	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> f Fn suc A )
17	7	sucid 5804	. . . . . . . . . . 11 |- A e. suc A
18		fnsnfv 6258	. . . . . . . . . . 11 |- ( ( f Fn suc A /\ A e. suc A ) -> { ( f ` A ) } = ( f " { A } ) )
19	16, 17, 18	sylancl 694	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> { ( f ` A ) } = ( f " { A } ) )
20	19	difeq2d 3728	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( ( f " suc A ) \ { ( f ` A ) } ) = ( ( f " suc A ) \ ( f " { A } ) ) )
21		imadmrn 5476	. . . . . . . . . . . 12 |- ( f " dom f ) = ran f
22	21	eqcomi 2631	. . . . . . . . . . 11 |- ran f = ( f " dom f )
23		f1ofo 6144	. . . . . . . . . . . 12 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -onto-> suc B )
24		forn 6118	. . . . . . . . . . . 12 |- ( f : suc A -onto-> suc B -> ran f = suc B )
25	23, 24	syl 17	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> ran f = suc B )
26		f1odm 6141	. . . . . . . . . . . 12 |- ( f : suc A -1-1-onto-> suc B -> dom f = suc A )
27	26	imaeq2d 5466	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> ( f " dom f ) = ( f " suc A ) )
28	22, 25, 27	3eqtr3a 2680	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> suc B = ( f " suc A ) )
29	28	difeq1d 3727	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( suc B \ { ( f ` A ) } ) = ( ( f " suc A ) \ { ( f ` A ) } ) )
30		dff1o3 6143	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B <-> ( f : suc A -onto-> suc B /\ Fun `' f ) )
31	30	simprbi 480	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> Fun `' f )
32		imadif 5973	. . . . . . . . . 10 |- ( Fun `' f -> ( f " ( suc A \ { A } ) ) = ( ( f " suc A ) \ ( f " { A } ) ) )
33	31, 32	syl 17	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( f " ( suc A \ { A } ) ) = ( ( f " suc A ) \ ( f " { A } ) ) )
34	20, 29, 33	3eqtr4rd 2667	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( f " ( suc A \ { A } ) ) = ( suc B \ { ( f ` A ) } ) )
35	15, 34	sylan9eq 2676	. . . . . . 7 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) = ( suc B \ { ( f ` A ) } ) )
36	11, 35	breqtrd 4679	. . . . . 6 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( suc B \ { ( f ` A ) } ) )
37		fnfvelrn 6356	. . . . . . . . . 10 |- ( ( f Fn suc A /\ A e. suc A ) -> ( f ` A ) e. ran f )
38	16, 17, 37	sylancl 694	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( f ` A ) e. ran f )
39	24	eleq2d 2687	. . . . . . . . . 10 |- ( f : suc A -onto-> suc B -> ( ( f ` A ) e. ran f <-> ( f ` A ) e. suc B ) )
40	23, 39	syl 17	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( ( f ` A ) e. ran f <-> ( f ` A ) e. suc B ) )
41	38, 40	mpbid 222	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( f ` A ) e. suc B )
42		fvex 6201	. . . . . . . . 9 |- ( f ` A ) e. _V
43	4, 42	phplem3 8141	. . . . . . . 8 |- ( ( B e. om /\ ( f ` A ) e. suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
44	41, 43	sylan2 491	. . . . . . 7 |- ( ( B e. om /\ f : suc A -1-1-onto-> suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
45	44	ensymd 8007	. . . . . 6 |- ( ( B e. om /\ f : suc A -1-1-onto-> suc B ) -> ( suc B \ { ( f ` A ) } ) ~~ B )
46		entr 8008	. . . . . 6 |- ( ( A ~~ ( suc B \ { ( f ` A ) } ) /\ ( suc B \ { ( f ` A ) } ) ~~ B ) -> A ~~ B )
47	36, 45, 46	syl2an 494	. . . . 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 )
48	47	anandirs 874	. . . 4 |- ( ( ( A e. om /\ B e. om ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ B )
49	48	ex 450	. . 3 |- ( ( A e. om /\ B e. om ) -> ( f : suc A -1-1-onto-> suc B -> A ~~ B ) )
50	49	exlimdv 1861	. 2 |- ( ( A e. om /\ B e. om ) -> ( E. f f : suc A -1-1-onto-> suc B -> A ~~ B ) )
51	1, 50	syl5bi 232	1 |- ( ( A e. om /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Ord word 5722   suc csuc 5725   Fun wfun 5882   Fn wfn 5883   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   omcom 7065   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-er 7742  df-en 7956

This theorem is referenced by:  nneneq  8143