Theorem phplem2 7038

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 4321	. . . . . . 7 |- <. B , A >. e. _V
4	3	snex 4275	. . . . . 6 |- { <. B , A >. } e. _V
5	1, 2	f1osn 5625	. . . . . 6 |- { <. B , A >. } : { B } -1-1-onto-> { A }
6		f1oen3g 6926	. . . . . 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 3333	. . . . . . 7 |- ( A \ { B } ) C_ A
9	2, 8	ssexi 4227	. . . . . 6 |- ( A \ { B } ) e. _V
10	9	enref 6937	. . . . 5 |- ( A \ { B } ) ~~ ( A \ { B } )
11	7, 10	pm3.2i 272	. . . 4 |- ( { B } ~~ { A } /\ ( A \ { B } ) ~~ ( A \ { B } ) )
12		incom 3399	. . . . . 6 |- ( { A } i^i ( A \ { B } ) ) = ( ( A \ { B } ) i^i { A } )
13		ssrin 3432	. . . . . . . . 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 4710	. . . . . . . . 9 |- ( A e. om -> Ord A )
16		orddisj 4644	. . . . . . . . 9 |- ( Ord A -> ( A i^i { A } ) = (/) )
17	15, 16	syl 14	. . . . . . . 8 |- ( A e. om -> ( A i^i { A } ) = (/) )
18	14, 17	sseqtrid 3277	. . . . . . 7 |- ( A e. om -> ( ( A \ { B } ) i^i { A } ) C_ (/) )
19		ss0 3535	. . . . . . 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 2276	. . . . 5 |- ( A e. om -> ( { A } i^i ( A \ { B } ) ) = (/) )
22		disjdif 3567	. . . . 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 6990	. . . 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 3351	. . 3 |- ( { B } u. ( A \ { B } ) ) = ( ( A \ { B } ) u. { B } )
28		nndifsnid 6674	. . 3 |- ( ( A e. om /\ B e. A ) -> ( ( A \ { B } ) u. { B } ) = A )
29	27, 28	eqtrid 2276	. 2 |- ( ( A e. om /\ B e. A ) -> ( { B } u. ( A \ { B } ) ) = A )
30		phplem1 7037	. 2 |- ( ( A e. om /\ B e. A ) -> ( { A } u. ( A \ { B } ) ) = ( suc A \ { B } ) )
31	26, 29, 30	3brtr3d 4119	1 |- ( ( A e. om /\ B e. A ) -> A ~~ ( suc A \ { B } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   \ cdif 3197   u. cun 3198   i^i cin 3199   C_ wss 3200   (/)c0 3494   {csn 3669   <.cop 3672   class class class wbr 4088   Ord word 4459   suc csuc 4462   omcom 4688   -1-1-onto->wf1o 5325   ~~ cen 6906

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-en 6909

This theorem is referenced by:  phplem3  7039