Theorem dif1en 7755

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 6705	. . . . 5 |- ( M e. om -> suc M e. om )
2		breq2 4441	. . . . . . 7 |- ( x = suc M -> ( A ~~ x <-> A ~~ suc M ) )
3	2	rspcev 3196	. . . . . 6 |- ( ( suc M e. om /\ A ~~ suc M ) -> E. x e. om A ~~ x )
4		isfi 7541	. . . . . 6 |- ( A e. Fin <-> E. x e. om A ~~ x )
5	3, 4	sylibr 212	. . . . 5 |- ( ( suc M e. om /\ A ~~ suc M ) -> A e. Fin )
6	1, 5	sylan 471	. . . 4 |- ( ( M e. om /\ A ~~ suc M ) -> A e. Fin )
7		diffi 7753	. . . . 5 |- ( A e. Fin -> ( A \ { X } ) e. Fin )
8		isfi 7541	. . . . 5 |- ( ( A \ { X } ) e. Fin <-> E. x e. om ( A \ { X } ) ~~ x )
9	7, 8	sylib 196	. . . 4 |- ( A e. Fin -> E. x e. om ( A \ { X } ) ~~ x )
10	6, 9	syl 16	. . 3 |- ( ( M e. om /\ A ~~ suc M ) -> E. x e. om ( A \ { X } ) ~~ x )
11	10	3adant3 1017	. 2 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> E. x e. om ( A \ { X } ) ~~ x )
12		vex 3098	. . . . . . . 8 |- x e. _V
13		en2sn 7597	. . . . . . . 8 |- ( ( X e. A /\ x e. _V ) -> { X } ~~ { x } )
14	12, 13	mpan2 671	. . . . . . 7 |- ( X e. A -> { X } ~~ { x } )
15		nnord 6693	. . . . . . . 8 |- ( x e. om -> Ord x )
16		orddisj 4906	. . . . . . . 8 |- ( Ord x -> ( x i^i { x } ) = (/) )
17	15, 16	syl 16	. . . . . . 7 |- ( x e. om -> ( x i^i { x } ) = (/) )
18		incom 3676	. . . . . . . . . 10 |- ( ( A \ { X } ) i^i { X } ) = ( { X } i^i ( A \ { X } ) )
19		disjdif 3886	. . . . . . . . . 10 |- ( { X } i^i ( A \ { X } ) ) = (/)
20	18, 19	eqtri 2472	. . . . . . . . 9 |- ( ( A \ { X } ) i^i { X } ) = (/)
21		unen 7600	. . . . . . . . . 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 826	. . . . . . . . 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 681	. . . . . . . 8 |- ( ( ( A \ { X } ) ~~ x /\ ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) ) -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) )
24	23	expcom 435	. . . . . . 7 |- ( ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
25	14, 17, 24	syl2an 477	. . . . . 6 |- ( ( X e. A /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
26	25	3ad2antl3 1161	. . . . 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 4161	. . . . . . . . 9 |- ( X e. A -> ( ( A \ { X } ) u. { X } ) = A )
28		df-suc 4874	. . . . . . . . . . 11 |- suc x = ( x u. { x } )
29	28	eqcomi 2456	. . . . . . . . . 10 |- ( x u. { x } ) = suc x
30	29	a1i 11	. . . . . . . . 9 |- ( X e. A -> ( x u. { x } ) = suc x )
31	27, 30	breq12d 4450	. . . . . . . 8 |- ( X e. A -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
32	31	3ad2ant3 1020	. . . . . . 7 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
33	32	adantr 465	. . . . . 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 7566	. . . . . . . . . . 11 |- ( A ~~ suc M -> suc M ~~ A )
35		entr 7569	. . . . . . . . . . . . 13 |- ( ( suc M ~~ A /\ A ~~ suc x ) -> suc M ~~ suc x )
36		peano2 6705	. . . . . . . . . . . . . 14 |- ( x e. om -> suc x e. om )
37		nneneq 7702	. . . . . . . . . . . . . 14 |- ( ( suc M e. om /\ suc x e. om ) -> ( suc M ~~ suc x <-> suc M = suc x ) )
38	36, 37	sylan2 474	. . . . . . . . . . . . 13 |- ( ( suc M e. om /\ x e. om ) -> ( suc M ~~ suc x <-> suc M = suc x ) )
39	35, 38	syl5ib 219	. . . . . . . . . . . 12 |- ( ( suc M e. om /\ x e. om ) -> ( ( suc M ~~ A /\ A ~~ suc x ) -> suc M = suc x ) )
40	39	expd 436	. . . . . . . . . . 11 |- ( ( suc M e. om /\ x e. om ) -> ( suc M ~~ A -> ( A ~~ suc x -> suc M = suc x ) ) )
41	34, 40	syl5 32	. . . . . . . . . 10 |- ( ( suc M e. om /\ x e. om ) -> ( A ~~ suc M -> ( A ~~ suc x -> suc M = suc x ) ) )
42	1, 41	sylan 471	. . . . . . . . 9 |- ( ( M e. om /\ x e. om ) -> ( A ~~ suc M -> ( A ~~ suc x -> suc M = suc x ) ) )
43	42	imp 429	. . . . . . . 8 |- ( ( ( M e. om /\ x e. om ) /\ A ~~ suc M ) -> ( A ~~ suc x -> suc M = suc x ) )
44	43	an32s 804	. . . . . . 7 |- ( ( ( M e. om /\ A ~~ suc M ) /\ x e. om ) -> ( A ~~ suc x -> suc M = suc x ) )
45	44	3adantl3 1155	. . . . . 6 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( A ~~ suc x -> suc M = suc x ) )
46	33, 45	sylbid 215	. . . . 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 6707	. . . . . . 7 |- ( ( M e. om /\ x e. om ) -> ( suc M = suc x <-> M = x ) )
48	47	biimpd 207	. . . . . 6 |- ( ( M e. om /\ x e. om ) -> ( suc M = suc x -> M = x ) )
49	48	3ad2antl1 1159	. . . . 5 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( suc M = suc x -> M = x ) )
50	26, 46, 49	3syld 55	. . . 4 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> M = x ) )
51		breq2 4441	. . . . 5 |- ( M = x -> ( ( A \ { X } ) ~~ M <-> ( A \ { X } ) ~~ x ) )
52	51	biimprcd 225	. . . 4 |- ( ( A \ { X } ) ~~ x -> ( M = x -> ( A \ { X } ) ~~ M ) )
53	50, 52	sylcom 29	. . 3 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> ( A \ { X } ) ~~ M ) )
54	53	rexlimdva 2935	. 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 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   E.wrex 2794   _Vcvv 3095   \ cdif 3458   u. cun 3459   i^i cin 3460   (/)c0 3770   {csn 4014   class class class wbr 4437   Ord word 4867   suc csuc 4870   omcom 6685   ~~ cen 7515   Fincfn 7518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-1o 7132  df-er 7313  df-en 7519  df-fin 7522

This theorem is referenced by:  enp1i  7757  findcard  7761  findcard2  7762  en2eleq  8389  en2other2  8390  mreexexlem4d  15025  f1otrspeq  16450  pmtrf  16458  pmtrmvd  16459  pmtrfinv  16464