Theorem dif1en 7808

Description: If a set A is equinumerous to the successor of a natural number M, then A with an element removed is equinumerous to M. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)

Assertion

Ref	Expression
dif1en	|- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( A \ { X } ) ~~ M )

Proof of Theorem dif1en

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2 6725	. . . . 5 |- ( M e. om -> suc M e. om )
2		breq2 4425	. . . . . . 7 |- ( x = suc M -> ( A ~~ x <-> A ~~ suc M ) )
3	2	rspcev 3183	. . . . . 6 |- ( ( suc M e. om /\ A ~~ suc M ) -> E. x e. om A ~~ x )
4		isfi 7598	. . . . . 6 |- ( A e. Fin <-> E. x e. om A ~~ x )
5	3, 4	sylibr 216	. . . . 5 |- ( ( suc M e. om /\ A ~~ suc M ) -> A e. Fin )
6	1, 5	sylan 474	. . . 4 |- ( ( M e. om /\ A ~~ suc M ) -> A e. Fin )
7		diffi 7807	. . . . 5 |- ( A e. Fin -> ( A \ { X } ) e. Fin )
8		isfi 7598	. . . . 5 |- ( ( A \ { X } ) e. Fin <-> E. x e. om ( A \ { X } ) ~~ x )
9	7, 8	sylib 200	. . . 4 |- ( A e. Fin -> E. x e. om ( A \ { X } ) ~~ x )
10	6, 9	syl 17	. . 3 |- ( ( M e. om /\ A ~~ suc M ) -> E. x e. om ( A \ { X } ) ~~ x )
11	10	3adant3 1026	. 2 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> E. x e. om ( A \ { X } ) ~~ x )
12		vex 3085	. . . . . . . 8 |- x e. _V
13		en2sn 7654	. . . . . . . 8 |- ( ( X e. A /\ x e. _V ) -> { X } ~~ { x } )
14	12, 13	mpan2 676	. . . . . . 7 |- ( X e. A -> { X } ~~ { x } )
15		nnord 6712	. . . . . . . 8 |- ( x e. om -> Ord x )
16		orddisj 5478	. . . . . . . 8 |- ( Ord x -> ( x i^i { x } ) = (/) )
17	15, 16	syl 17	. . . . . . 7 |- ( x e. om -> ( x i^i { x } ) = (/) )
18		incom 3656	. . . . . . . . . 10 |- ( ( A \ { X } ) i^i { X } ) = ( { X } i^i ( A \ { X } ) )
19		disjdif 3868	. . . . . . . . . 10 |- ( { X } i^i ( A \ { X } ) ) = (/)
20	18, 19	eqtri 2452	. . . . . . . . 9 |- ( ( A \ { X } ) i^i { X } ) = (/)
21		unen 7657	. . . . . . . . . 10 |- ( ( ( ( A \ { X } ) ~~ x /\ { X } ~~ { x } ) /\ ( ( ( A \ { X } ) i^i { X } ) = (/) /\ ( x i^i { x } ) = (/) ) ) -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) )
22	21	an4s 834	. . . . . . . . 9 |- ( ( ( ( A \ { X } ) ~~ x /\ ( ( A \ { X } ) i^i { X } ) = (/) ) /\ ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) ) -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) )
23	20, 22	mpanl2 686	. . . . . . . 8 |- ( ( ( A \ { X } ) ~~ x /\ ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) ) -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) )
24	23	expcom 437	. . . . . . 7 |- ( ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
25	14, 17, 24	syl2an 480	. . . . . 6 |- ( ( X e. A /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
26	25	3ad2antl3 1170	. . . . 5 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
27		difsnid 4144	. . . . . . . . 9 |- ( X e. A -> ( ( A \ { X } ) u. { X } ) = A )
28		df-suc 5446	. . . . . . . . . . 11 |- suc x = ( x u. { x } )
29	28	eqcomi 2436	. . . . . . . . . 10 |- ( x u. { x } ) = suc x
30	29	a1i 11	. . . . . . . . 9 |- ( X e. A -> ( x u. { x } ) = suc x )
31	27, 30	breq12d 4434	. . . . . . . 8 |- ( X e. A -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
32	31	3ad2ant3 1029	. . . . . . 7 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
33	32	adantr 467	. . . . . 6 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
34		ensym 7623	. . . . . . . . . . 11 |- ( A ~~ suc M -> suc M ~~ A )
35		entr 7626	. . . . . . . . . . . . 13 |- ( ( suc M ~~ A /\ A ~~ suc x ) -> suc M ~~ suc x )
36		peano2 6725	. . . . . . . . . . . . . 14 |- ( x e. om -> suc x e. om )
37		nneneq 7759	. . . . . . . . . . . . . 14 |- ( ( suc M e. om /\ suc x e. om ) -> ( suc M ~~ suc x <-> suc M = suc x ) )
38	36, 37	sylan2 477	. . . . . . . . . . . . 13 |- ( ( suc M e. om /\ x e. om ) -> ( suc M ~~ suc x <-> suc M = suc x ) )
39	35, 38	syl5ib 223	. . . . . . . . . . . 12 |- ( ( suc M e. om /\ x e. om ) -> ( ( suc M ~~ A /\ A ~~ suc x ) -> suc M = suc x ) )
40	39	expd 438	. . . . . . . . . . 11 |- ( ( suc M e. om /\ x e. om ) -> ( suc M ~~ A -> ( A ~~ suc x -> suc M = suc x ) ) )
41	34, 40	syl5 34	. . . . . . . . . 10 |- ( ( suc M e. om /\ x e. om ) -> ( A ~~ suc M -> ( A ~~ suc x -> suc M = suc x ) ) )
42	1, 41	sylan 474	. . . . . . . . 9 |- ( ( M e. om /\ x e. om ) -> ( A ~~ suc M -> ( A ~~ suc x -> suc M = suc x ) ) )
43	42	imp 431	. . . . . . . 8 |- ( ( ( M e. om /\ x e. om ) /\ A ~~ suc M ) -> ( A ~~ suc x -> suc M = suc x ) )
44	43	an32s 812	. . . . . . 7 |- ( ( ( M e. om /\ A ~~ suc M ) /\ x e. om ) -> ( A ~~ suc x -> suc M = suc x ) )
45	44	3adantl3 1164	. . . . . 6 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( A ~~ suc x -> suc M = suc x ) )
46	33, 45	sylbid 219	. . . . 5 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) -> suc M = suc x ) )
47		peano4 6727	. . . . . . 7 |- ( ( M e. om /\ x e. om ) -> ( suc M = suc x <-> M = x ) )
48	47	biimpd 211	. . . . . 6 |- ( ( M e. om /\ x e. om ) -> ( suc M = suc x -> M = x ) )
49	48	3ad2antl1 1168	. . . . 5 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( suc M = suc x -> M = x ) )
50	26, 46, 49	3syld 58	. . . 4 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> M = x ) )
51		breq2 4425	. . . . 5 |- ( M = x -> ( ( A \ { X } ) ~~ M <-> ( A \ { X } ) ~~ x ) )
52	51	biimprcd 229	. . . 4 |- ( ( A \ { X } ) ~~ x -> ( M = x -> ( A \ { X } ) ~~ M ) )
53	50, 52	sylcom 31	. . 3 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> ( A \ { X } ) ~~ M ) )
54	53	rexlimdva 2918	. 2 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( E. x e. om ( A \ { X } ) ~~ x -> ( A \ { X } ) ~~ M ) )
55	11, 54	mpd 15	1 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( A \ { X } ) ~~ M )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 983   = wceq 1438   e. wcel 1869   E.wrex 2777   _Vcvv 3082   \ cdif 3434   u. cun 3435   i^i cin 3436   (/)c0 3762   {csn 3997   class class class wbr 4421   Ord word 5439   suc csuc 5442   omcom 6704   ~~ cen 7572   Fincfn 7575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-om 6705  df-1o 7188  df-er 7369  df-en 7576  df-fin 7579

This theorem is referenced by:  enp1i  7810  findcard  7814  findcard2  7815  en2eleq  8442  en2other2  8443  mreexexlem4d  15546  f1otrspeq  17081  pmtrf  17089  pmtrmvd  17090  pmtrfinv  17095