Theorem phplem4 6830

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 6722	. 2 |- ( suc A ~~ suc B <-> E. f f : suc A -1-1-onto-> suc B )
2		f1of1 5439	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -1-1-> suc B )
3	2	adantl 275	. . . . . . . . 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 4481	. . . . . . . . 9 |- suc B e. _V
6		sssucid 4398	. . . . . . . . . 10 |- A C_ suc A
7		phplem2.1	. . . . . . . . . 10 |- A e. _V
8		f1imaen2g 6768	. . . . . . . . . 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 437	. . . . . . . . 9 |- ( ( f : suc A -1-1-> suc B /\ suc B e. _V ) -> ( f " A ) ~~ A )
10	3, 5, 9	sylancl 411	. . . . . . . 8 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) ~~ A )
11	10	ensymd 6758	. . . . . . 7 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( f " A ) )
12		nnord 4594	. . . . . . . . . 10 |- ( A e. om -> Ord A )
13		orddif 4529	. . . . . . . . . 10 |- ( Ord A -> A = ( suc A \ { A } ) )
14	12, 13	syl 14	. . . . . . . . 9 |- ( A e. om -> A = ( suc A \ { A } ) )
15	14	imaeq2d 4951	. . . . . . . 8 |- ( A e. om -> ( f " A ) = ( f " ( suc A \ { A } ) ) )
16		f1ofn 5441	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> f Fn suc A )
17	7	sucid 4400	. . . . . . . . . . 11 |- A e. suc A
18		fnsnfv 5553	. . . . . . . . . . 11 |- ( ( f Fn suc A /\ A e. suc A ) -> { ( f ` A ) } = ( f " { A } ) )
19	16, 17, 18	sylancl 411	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> { ( f ` A ) } = ( f " { A } ) )
20	19	difeq2d 3245	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( ( f " suc A ) \ { ( f ` A ) } ) = ( ( f " suc A ) \ ( f " { A } ) ) )
21		imadmrn 4961	. . . . . . . . . . . 12 |- ( f " dom f ) = ran f
22	21	eqcomi 2174	. . . . . . . . . . 11 |- ran f = ( f " dom f )
23		f1ofo 5447	. . . . . . . . . . . 12 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -onto-> suc B )
24		forn 5421	. . . . . . . . . . . 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 5444	. . . . . . . . . . . 12 |- ( f : suc A -1-1-onto-> suc B -> dom f = suc A )
27	26	imaeq2d 4951	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> ( f " dom f ) = ( f " suc A ) )
28	22, 25, 27	3eqtr3a 2227	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> suc B = ( f " suc A ) )
29	28	difeq1d 3244	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( suc B \ { ( f ` A ) } ) = ( ( f " suc A ) \ { ( f ` A ) } ) )
30		dff1o3 5446	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B <-> ( f : suc A -onto-> suc B /\ Fun `' f ) )
31	30	simprbi 273	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> Fun `' f )
32		imadif 5276	. . . . . . . . . 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 2214	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( f " ( suc A \ { A } ) ) = ( suc B \ { ( f ` A ) } ) )
35	15, 34	sylan9eq 2223	. . . . . . 7 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) = ( suc B \ { ( f ` A ) } ) )
36	11, 35	breqtrd 4013	. . . . . 6 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( suc B \ { ( f ` A ) } ) )
37		fnfvelrn 5626	. . . . . . . . . 10 |- ( ( f Fn suc A /\ A e. suc A ) -> ( f ` A ) e. ran f )
38	16, 17, 37	sylancl 411	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( f ` A ) e. ran f )
39	24	eleq2d 2240	. . . . . . . . . 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 146	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( f ` A ) e. suc B )
42		vex 2733	. . . . . . . . . 10 |- f e. _V
43	42, 7	fvex 5514	. . . . . . . . 9 |- ( f ` A ) e. _V
44	4, 43	phplem3 6829	. . . . . . . 8 |- ( ( B e. om /\ ( f ` A ) e. suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
45	41, 44	sylan2 284	. . . . . . 7 |- ( ( B e. om /\ f : suc A -1-1-onto-> suc B ) -> B ~~ ( suc B \ { ( f ` A ) } ) )
46	45	ensymd 6758	. . . . . 6 |- ( ( B e. om /\ f : suc A -1-1-onto-> suc B ) -> ( suc B \ { ( f ` A ) } ) ~~ B )
47		entr 6759	. . . . . 6 |- ( ( A ~~ ( suc B \ { ( f ` A ) } ) /\ ( suc B \ { ( f ` A ) } ) ~~ B ) -> A ~~ B )
48	36, 46, 47	syl2an 287	. . . . 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 588	. . . 4 |- ( ( ( A e. om /\ B e. om ) /\ f : suc A -1-1-onto-> suc B ) -> A ~~ B )
50	49	ex 114	. . 3 |- ( ( A e. om /\ B e. om ) -> ( f : suc A -1-1-onto-> suc B -> A ~~ B ) )
51	50	exlimdv 1812	. 2 |- ( ( A e. om /\ B e. om ) -> ( E. f f : suc A -1-1-onto-> suc B -> A ~~ B ) )
52	1, 51	syl5bi 151	1 |- ( ( A e. om /\ B e. om ) -> ( suc A ~~ suc B -> A ~~ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   \ cdif 3118   C_ wss 3121   {csn 3581   class class class wbr 3987   Ord word 4345   suc csuc 4348   omcom 4572   `'ccnv 4608   dom cdm 4609   ran crn 4610   "cima 4612   Fun wfun 5190   Fn wfn 5191   -1-1->wf1 5193   -onto->wfo 5194   -1-1-onto->wf1o 5195   ` cfv 5196   ~~ cen 6713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-er 6510  df-en 6716

This theorem is referenced by:  nneneq  6832  php5  6833