Theorem phplem2 7690

Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)

Hypotheses

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

Assertion

Ref	Expression
phplem2	|- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )

Proof of Theorem phplem2

Step	Hyp	Ref	Expression
1		snex 4678	. . . . . 6 |- { <. B , A >. } e. _V
2		phplem2.2	. . . . . . 7 |- B e. _V
3		phplem2.1	. . . . . . 7 |- A e. _V
4	2, 3	f1osn 5835	. . . . . 6 |- { <. B , A >. } : { B } -1-1-onto-> { A }
5		f1oen3g 7524	. . . . . 6 |- ( ( { <. B , A >. } e. _V /\ { <. B , A >. } : { B } -1-1-onto-> { A } ) -> { B } ~~ { A } )
6	1, 4, 5	mp2an 670	. . . . 5 |- { B } ~~ { A }
7		difss 3617	. . . . . . 7 |- ( A \ { B } ) C_ A
8	3, 7	ssexi 4582	. . . . . 6 |- ( A \ { B } ) e. _V
9	8	enref 7541	. . . . 5 |- ( A \ { B } ) ~~ ( A \ { B } )
10	6, 9	pm3.2i 453	. . . 4 |- ( { B } ~~ { A } /\ ( A \ { B } ) ~~ ( A \ { B } ) )
11		incom 3677	. . . . . 6 |- ( { A } i^i ( A \ { B } ) ) = ( ( A \ { B } ) i^i { A } )
12		ssrin 3709	. . . . . . . . 9 |- ( ( A \ { B } ) C_ A -> ( ( A \ { B } ) i^i { A } ) C_ ( A i^i { A } ) )
13	7, 12	ax-mp 5	. . . . . . . 8 |- ( ( A \ { B } ) i^i { A } ) C_ ( A i^i { A } )
14		nnord 6681	. . . . . . . . 9 |- ( A e. om -> Ord A )
15		orddisj 4905	. . . . . . . . 9 |- ( Ord A -> ( A i^i { A } ) = (/) )
16	14, 15	syl 16	. . . . . . . 8 |- ( A e. om -> ( A i^i { A } ) = (/) )
17	13, 16	syl5sseq 3537	. . . . . . 7 |- ( A e. om -> ( ( A \ { B } ) i^i { A } ) C_ (/) )
18		ss0 3815	. . . . . . 7 |- ( ( ( A \ { B } ) i^i { A } ) C_ (/) -> ( ( A \ { B } ) i^i { A } ) = (/) )
19	17, 18	syl 16	. . . . . 6 |- ( A e. om -> ( ( A \ { B } ) i^i { A } ) = (/) )
20	11, 19	syl5eq 2507	. . . . 5 |- ( A e. om -> ( { A } i^i ( A \ { B } ) ) = (/) )
21		disjdif 3888	. . . . 5 |- ( { B } i^i ( A \ { B } ) ) = (/)
22	20, 21	jctil 535	. . . 4 |- ( A e. om -> ( ( { B } i^i ( A \ { B } ) ) = (/) /\ ( { A } i^i ( A \ { B } ) ) = (/) ) )
23		unen 7591	. . . 4 |- ( ( ( { B } ~~ { A } /\ ( A \ { B } ) ~~ ( A \ { B } ) ) /\ ( ( { B } i^i ( A \ { B } ) ) = (/) /\ ( { A } i^i ( A \ { B } ) ) = (/) ) ) -> ( { B } u. ( A \ { B } ) ) ~~ ( { A } u. ( A \ { B } ) ) )
24	10, 22, 23	sylancr 661	. . 3 |- ( A e. om -> ( { B } u. ( A \ { B } ) ) ~~ ( { A } u. ( A \ { B } ) ) )
25	24	adantr 463	. 2 |- ( ( A e. om /\ B e. A ) -> ( { B } u. ( A \ { B } ) ) ~~ ( { A } u. ( A \ { B } ) ) )
26		uncom 3634	. . . 4 |- ( { B } u. ( A \ { B } ) ) = ( ( A \ { B } ) u. { B } )
27		difsnid 4162	. . . 4 |- ( B e. A -> ( ( A \ { B } ) u. { B } ) = A )
28	26, 27	syl5eq 2507	. . 3 |- ( B e. A -> ( { B } u. ( A \ { B } ) ) = A )
29	28	adantl 464	. 2 |- ( ( A e. om /\ B e. A ) -> ( { B } u. ( A \ { B } ) ) = A )
30		phplem1 7689	. 2 |- ( ( A e. om /\ B e. A ) -> ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) )
31	25, 29, 30	3brtr3d 4468	1 |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   \ cdif 3458   u. cun 3459   i^i cin 3460   C_ wss 3461   (/)c0 3783   {csn 4016   <.cop 4022   class class class wbr 4439   Ord word 4866   suc csuc 4869   -1-1-onto->wf1o 5569   omcom 6673   ~~ cen 7506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-om 6674  df-en 7510

This theorem is referenced by:  phplem3  7691