Theorem phplem2 7034

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		phplem2.2	. . . . . . . 8 |- B e. _V
2		phplem2.1	. . . . . . . 8 |- A e. _V
3	1, 2	opex 4319	. . . . . . 7 |- <. B , A >. e. _V
4	3	snex 4273	. . . . . 6 |- { <. B , A >. } e. _V
5	1, 2	f1osn 5621	. . . . . 6 |- { <. B , A >. } : { B } -1-1-onto-> { A }
6		f1oen3g 6922	. . . . . 6 |- ( ( { <. B , A >. } e. _V /\ { <. B , A >. } : { B } -1-1-onto-> { A } ) -> { B } ~~ { A } )
7	4, 5, 6	mp2an 426	. . . . 5 |- { B } ~~ { A }
8		difss 3331	. . . . . . 7 |- ( A \ { B } ) C_ A
9	2, 8	ssexi 4225	. . . . . 6 |- ( A \ { B } ) e. _V
10	9	enref 6933	. . . . 5 |- ( A \ { B } ) ~~ ( A \ { B } )
11	7, 10	pm3.2i 272	. . . 4 |- ( { B } ~~ { A } /\ ( A \ { B } ) ~~ ( A \ { B } ) )
12		incom 3397	. . . . . 6 |- ( { A } i^i ( A \ { B } ) ) = ( ( A \ { B } ) i^i { A } )
13		ssrin 3430	. . . . . . . . 9 |- ( ( A \ { B } ) C_ A -> ( ( A \ { B } ) i^i { A } ) C_ ( A i^i { A } ) )
14	8, 13	ax-mp 5	. . . . . . . 8 |- ( ( A \ { B } ) i^i { A } ) C_ ( A i^i { A } )
15		nnord 4708	. . . . . . . . 9 |- ( A e. om -> Ord A )
16		orddisj 4642	. . . . . . . . 9 |- ( Ord A -> ( A i^i { A } ) = (/) )
17	15, 16	syl 14	. . . . . . . 8 |- ( A e. om -> ( A i^i { A } ) = (/) )
18	14, 17	sseqtrid 3275	. . . . . . 7 |- ( A e. om -> ( ( A \ { B } ) i^i { A } ) C_ (/) )
19		ss0 3533	. . . . . . 7 |- ( ( ( A \ { B } ) i^i { A } ) C_ (/) -> ( ( A \ { B } ) i^i { A } ) = (/) )
20	18, 19	syl 14	. . . . . 6 |- ( A e. om -> ( ( A \ { B } ) i^i { A } ) = (/) )
21	12, 20	eqtrid 2274	. . . . 5 |- ( A e. om -> ( { A } i^i ( A \ { B } ) ) = (/) )
22		disjdif 3565	. . . . 5 |- ( { B } i^i ( A \ { B } ) ) = (/)
23	21, 22	jctil 312	. . . 4 |- ( A e. om -> ( ( { B } i^i ( A \ { B } ) ) = (/) /\ ( { A } i^i ( A \ { B } ) ) = (/) ) )
24		unen 6986	. . . 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 } ) ) )
25	11, 23, 24	sylancr 414	. . 3 |- ( A e. om -> ( { B } u. ( A \ { B } ) ) ~~ ( { A } u. ( A \ { B } ) ) )
26	25	adantr 276	. 2 |- ( ( A e. om /\ B e. A ) -> ( { B } u. ( A \ { B } ) ) ~~ ( { A } u. ( A \ { B } ) ) )
27		uncom 3349	. . 3 |- ( { B } u. ( A \ { B } ) ) = ( ( A \ { B } ) u. { B } )
28		nndifsnid 6670	. . 3 |- ( ( A e. om /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
29	27, 28	eqtrid 2274	. 2 |- ( ( A e. om /\ B e. A ) -> ( { B } u. ( A \ { B } ) ) = A )
30		phplem1 7033	. 2 |- ( ( A e. om /\ B e. A ) -> ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) )
31	26, 29, 30	3brtr3d 4117	1 |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   \ cdif 3195   u. cun 3196   i^i cin 3197   C_ wss 3198   (/)c0 3492   {csn 3667   <.cop 3670   class class class wbr 4086   Ord word 4457   suc csuc 4460   omcom 4686   -1-1-onto->wf1o 5323   ~~ cen 6902

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 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

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 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-en 6905

This theorem is referenced by:  phplem3  7035