Theorem phplem4 6972

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 6853	. 2 |- ( suc A ~~ suc B <-> E. f f : suc A -1-1-onto-> suc B )
2		f1of1 5538	. . . . . . . . . 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 4560	. . . . . . . . 9 |- suc B e. _V
6		sssucid 4475	. . . . . . . . . 10 |- A C_ suc A
7		phplem2.1	. . . . . . . . . 10 |- A e. _V
8		f1imaen2g 6903	. . . . . . . . . 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 6893	. . . . . . 7 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( f " A ) )
12		nnord 4673	. . . . . . . . . 10 |- ( A e. om -> Ord A )
13		orddif 4608	. . . . . . . . . 10 |- ( Ord A -> A = ( suc A \ { A } ) )
14	12, 13	syl 14	. . . . . . . . 9 |- ( A e. om -> A = ( suc A \ { A } ) )
15	14	imaeq2d 5036	. . . . . . . 8 |- ( A e. om -> ( f " A ) = ( f " ( suc A \ { A } ) ) )
16		f1ofn 5540	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> f Fn suc A )
17	7	sucid 4477	. . . . . . . . . . 11 |- A e. suc A
18		fnsnfv 5656	. . . . . . . . . . 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 3295	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( ( f " suc A ) \ { ( f ` A ) } ) = ( ( f " suc A ) \ ( f " { A } ) ) )
21		imadmrn 5046	. . . . . . . . . . . 12 |- ( f " dom f ) = ran f
22	21	eqcomi 2210	. . . . . . . . . . 11 |- ran f = ( f " dom f )
23		f1ofo 5546	. . . . . . . . . . . 12 |- ( f : suc A -1-1-onto-> suc B -> f : suc A -onto-> suc B )
24		forn 5518	. . . . . . . . . . . 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 5543	. . . . . . . . . . . 12 |- ( f : suc A -1-1-onto-> suc B -> dom f = suc A )
27	26	imaeq2d 5036	. . . . . . . . . . 11 |- ( f : suc A -1-1-onto-> suc B -> ( f " dom f ) = ( f " suc A ) )
28	22, 25, 27	3eqtr3a 2263	. . . . . . . . . 10 |- ( f : suc A -1-1-onto-> suc B -> suc B = ( f " suc A ) )
29	28	difeq1d 3294	. . . . . . . . 9 |- ( f : suc A -1-1-onto-> suc B -> ( suc B \ { ( f ` A ) } ) = ( ( f " suc A ) \ { ( f ` A ) } ) )
30		dff1o3 5545	. . . . . . . . . . 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 5368	. . . . . . . . . 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 2250	. . . . . . . 8 |- ( f : suc A -1-1-onto-> suc B -> ( f " ( suc A \ { A } ) ) = ( suc B \ { ( f ` A ) } ) )
35	15, 34	sylan9eq 2259	. . . . . . 7 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> ( f " A ) = ( suc B \ { ( f ` A ) } ) )
36	11, 35	breqtrd 4080	. . . . . 6 |- ( ( A e. om /\ f : suc A -1-1-onto-> suc B ) -> A ~~ ( suc B \ { ( f ` A ) } ) )
37		fnfvelrn 5730	. . . . . . . . . 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 2276	. . . . . . . . . 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 2776	. . . . . . . . . 10 |- f e. _V
43	42, 7	fvex 5614	. . . . . . . . 9 |- ( f ` A ) e. _V
44	4, 43	phplem3 6971	. . . . . . . 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 6893	. . . . . 6 |- ( ( B e. om /\ f : suc A -1-1-onto-> suc B ) -> ( suc B \ { ( f ` A ) } ) ~~ B )
47		entr 6894	. . . . . 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 593	. . . 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 1843	. 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 1373   E.wex 1516   e. wcel 2177   _Vcvv 2773   \ cdif 3167   C_ wss 3170   {csn 3638   class class class wbr 4054   Ord word 4422   suc csuc 4425   omcom 4651   `'ccnv 4687   dom cdm 4688   ran crn 4689   "cima 4691   Fun wfun 5279   Fn wfn 5280   -1-1->wf1 5282   -onto->wfo 5283   -1-1-onto->wf1o 5284   ` cfv 5285   ~~ cen 6843

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-er 6638  df-en 6846

This theorem is referenced by:  nneneq  6974  php5  6975