Theorem dif1en 7745

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 6693	. . . . 5 |- ( M e. om -> suc M e. om )
2		breq2 4443	. . . . . . 7 |- ( x = suc M -> ( A ~~ x <-> A ~~ suc M ) )
3	2	rspcev 3207	. . . . . 6 |- ( ( suc M e. om /\ A ~~ suc M ) -> E. x e. om A ~~ x )
4		isfi 7532	. . . . . 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 469	. . . 4 |- ( ( M e. om /\ A ~~ suc M ) -> A e. Fin )
7		diffi 7744	. . . . 5 |- ( A e. Fin -> ( A \ { X } ) e. Fin )
8		isfi 7532	. . . . 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 1014	. 2 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> E. x e. om ( A \ { X } ) ~~ x )
12		vex 3109	. . . . . . . 8 |- x e. _V
13		en2sn 7588	. . . . . . . 8 |- ( ( X e. A /\ x e. _V ) -> { X } ~~ { x } )
14	12, 13	mpan2 669	. . . . . . 7 |- ( X e. A -> { X } ~~ { x } )
15		nnord 6681	. . . . . . . 8 |- ( x e. om -> Ord x )
16		orddisj 4905	. . . . . . . 8 |- ( Ord x -> ( x i^i { x } ) = (/) )
17	15, 16	syl 16	. . . . . . 7 |- ( x e. om -> ( x i^i { x } ) = (/) )
18		incom 3677	. . . . . . . . . 10 |- ( ( A \ { X } ) i^i { X } ) = ( { X } i^i ( A \ { X } ) )
19		disjdif 3888	. . . . . . . . . 10 |- ( { X } i^i ( A \ { X } ) ) = (/)
20	18, 19	eqtri 2483	. . . . . . . . 9 |- ( ( A \ { X } ) i^i { X } ) = (/)
21		unen 7591	. . . . . . . . . 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 824	. . . . . . . . 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 679	. . . . . . . 8 |- ( ( ( A \ { X } ) ~~ x /\ ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) ) -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) )
24	23	expcom 433	. . . . . . 7 |- ( ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
25	14, 17, 24	syl2an 475	. . . . . 6 |- ( ( X e. A /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
26	25	3ad2antl3 1158	. . . . 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 4162	. . . . . . . . 9 |- ( X e. A -> ( ( A \ { X } ) u. { X } ) = A )
28		df-suc 4873	. . . . . . . . . . 11 |- suc x = ( x u. { x } )
29	28	eqcomi 2467	. . . . . . . . . 10 |- ( x u. { x } ) = suc x
30	29	a1i 11	. . . . . . . . 9 |- ( X e. A -> ( x u. { x } ) = suc x )
31	27, 30	breq12d 4452	. . . . . . . 8 |- ( X e. A -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
32	31	3ad2ant3 1017	. . . . . . 7 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
33	32	adantr 463	. . . . . 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 7557	. . . . . . . . . . 11 |- ( A ~~ suc M -> suc M ~~ A )
35		entr 7560	. . . . . . . . . . . . 13 |- ( ( suc M ~~ A /\ A ~~ suc x ) -> suc M ~~ suc x )
36		peano2 6693	. . . . . . . . . . . . . 14 |- ( x e. om -> suc x e. om )
37		nneneq 7693	. . . . . . . . . . . . . 14 |- ( ( suc M e. om /\ suc x e. om ) -> ( suc M ~~ suc x <-> suc M = suc x ) )
38	36, 37	sylan2 472	. . . . . . . . . . . . 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 434	. . . . . . . . . . 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 469	. . . . . . . . 9 |- ( ( M e. om /\ x e. om ) -> ( A ~~ suc M -> ( A ~~ suc x -> suc M = suc x ) ) )
43	42	imp 427	. . . . . . . 8 |- ( ( ( M e. om /\ x e. om ) /\ A ~~ suc M ) -> ( A ~~ suc x -> suc M = suc x ) )
44	43	an32s 802	. . . . . . 7 |- ( ( ( M e. om /\ A ~~ suc M ) /\ x e. om ) -> ( A ~~ suc x -> suc M = suc x ) )
45	44	3adantl3 1152	. . . . . 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 6695	. . . . . . 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 1156	. . . . 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 4443	. . . . 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 2946	. 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 367   /\ w3a 971   = wceq 1398   e. wcel 1823   E.wrex 2805   _Vcvv 3106   \ cdif 3458   u. cun 3459   i^i cin 3460   (/)c0 3783   {csn 4016   class class class wbr 4439   Ord word 4866   suc csuc 4869   omcom 6673   ~~ cen 7506   Fincfn 7509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-er 7303  df-en 7510  df-fin 7513

This theorem is referenced by:  enp1i  7747  findcard  7751  findcard2  7752  en2eleq  8377  en2other2  8378  mreexexlem4d  15136  f1otrspeq  16671  pmtrf  16679  pmtrmvd  16680  pmtrfinv  16685