Theorem phplem2 7602

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 4642	. . . . . 6 |- { <. B , A >. } e. _V
2		phplem2.2	. . . . . . 7 |- B e. _V
3		phplem2.1	. . . . . . 7 |- A e. _V
4	2, 3	f1osn 5787	. . . . . 6 |- { <. B , A >. } : { B } -1-1-onto-> { A }
5		f1oen3g 7436	. . . . . 6 |- ( ( { <. B , A >. } e. _V /\ { <. B , A >. } : { B } -1-1-onto-> { A } ) -> { B } ~~ { A } )
6	1, 4, 5	mp2an 672	. . . . 5 |- { B } ~~ { A }
7		difss 3592	. . . . . . 7 |- ( A \ { B } ) C_ A
8	3, 7	ssexi 4546	. . . . . 6 |- ( A \ { B } ) e. _V
9	8	enref 7453	. . . . 5 |- ( A \ { B } ) ~~ ( A \ { B } )
10	6, 9	pm3.2i 455	. . . 4 |- ( { B } ~~ { A } /\ ( A \ { B } ) ~~ ( A \ { B } ) )
11		incom 3652	. . . . . 6 |- ( { A } i^i ( A \ { B } ) ) = ( ( A \ { B } ) i^i { A } )
12		ssrin 3684	. . . . . . . . 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 6595	. . . . . . . . 9 |- ( A e. om -> Ord A )
15		orddisj 4866	. . . . . . . . 9 |- ( Ord A -> ( A i^i { A } ) = (/) )
16	14, 15	syl 16	. . . . . . . 8 |- ( A e. om -> ( A i^i { A } ) = (/) )
17	13, 16	syl5sseq 3513	. . . . . . 7 |- ( A e. om -> ( ( A \ { B } ) i^i { A } ) C_ (/) )
18		ss0 3777	. . . . . . 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 3860	. . . . 5 |- ( { B } i^i ( A \ { B } ) ) = (/)
22	20, 21	jctil 537	. . . 4 |- ( A e. om -> ( ( { B } i^i ( A \ { B } ) ) = (/) /\ ( { A } i^i ( A \ { B } ) ) = (/) ) )
23		unen 7503	. . . 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 663	. . 3 |- ( A e. om -> ( { B } u. ( A \ { B } ) ) ~~ ( { A } u. ( A \ { B } ) ) )
25	24	adantr 465	. 2 |- ( ( A e. om /\ B e. A ) -> ( { B } u. ( A \ { B } ) ) ~~ ( { A } u. ( A \ { B } ) ) )
26		uncom 3609	. . . 4 |- ( { B } u. ( A \ { B } ) ) = ( ( A \ { B } ) u. { B } )
27		difsnid 4128	. . . 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 466	. 2 |- ( ( A e. om /\ B e. A ) -> ( { B } u. ( A \ { B } ) ) = A )
30		phplem1 7601	. 2 |- ( ( A e. om /\ B e. A ) -> ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) )
31	25, 29, 30	3brtr3d 4430	1 |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3078   \ cdif 3434   u. cun 3435   i^i cin 3436   C_ wss 3437   (/)c0 3746   {csn 3986   <.cop 3992   class class class wbr 4401   Ord word 4827   suc csuc 4830   -1-1-onto->wf1o 5526   omcom 6587   ~~ cen 7418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-om 6588  df-en 7422

This theorem is referenced by:  phplem3  7603