Theorem dif1en 7822

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 6732	. . . . 5 |- ( M e. om -> suc M e. om )
2		breq2 4399	. . . . . . 7 |- ( x = suc M -> ( A ~~ x <-> A ~~ suc M ) )
3	2	rspcev 3136	. . . . . 6 |- ( ( suc M e. om /\ A ~~ suc M ) -> E. x e. om A ~~ x )
4		isfi 7611	. . . . . 6 |- ( A e. Fin <-> E. x e. om A ~~ x )
5	3, 4	sylibr 217	. . . . 5 |- ( ( suc M e. om /\ A ~~ suc M ) -> A e. Fin )
6	1, 5	sylan 479	. . . 4 |- ( ( M e. om /\ A ~~ suc M ) -> A e. Fin )
7		diffi 7821	. . . . 5 |- ( A e. Fin -> ( A \ { X } ) e. Fin )
8		isfi 7611	. . . . 5 |- ( ( A \ { X } ) e. Fin <-> E. x e. om ( A \ { X } ) ~~ x )
9	7, 8	sylib 201	. . . 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 1050	. 2 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> E. x e. om ( A \ { X } ) ~~ x )
12		vex 3034	. . . . . . . 8 |- x e. _V
13		en2sn 7667	. . . . . . . 8 |- ( ( X e. A /\ x e. _V ) -> { X } ~~ { x } )
14	12, 13	mpan2 685	. . . . . . 7 |- ( X e. A -> { X } ~~ { x } )
15		nnord 6719	. . . . . . . 8 |- ( x e. om -> Ord x )
16		orddisj 5468	. . . . . . . 8 |- ( Ord x -> ( x i^i { x } ) = (/) )
17	15, 16	syl 17	. . . . . . 7 |- ( x e. om -> ( x i^i { x } ) = (/) )
18		incom 3616	. . . . . . . . . 10 |- ( ( A \ { X } ) i^i { X } ) = ( { X } i^i ( A \ { X } ) )
19		disjdif 3830	. . . . . . . . . 10 |- ( { X } i^i ( A \ { X } ) ) = (/)
20	18, 19	eqtri 2493	. . . . . . . . 9 |- ( ( A \ { X } ) i^i { X } ) = (/)
21		unen 7670	. . . . . . . . . 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 842	. . . . . . . . 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 695	. . . . . . . 8 |- ( ( ( A \ { X } ) ~~ x /\ ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) ) -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) )
24	23	expcom 442	. . . . . . 7 |- ( ( { X } ~~ { x } /\ ( x i^i { x } ) = (/) ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
25	14, 17, 24	syl2an 485	. . . . . 6 |- ( ( X e. A /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) ) )
26	25	3ad2antl3 1194	. . . . 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 4109	. . . . . . . . 9 |- ( X e. A -> ( ( A \ { X } ) u. { X } ) = A )
28		df-suc 5436	. . . . . . . . . . 11 |- suc x = ( x u. { x } )
29	28	eqcomi 2480	. . . . . . . . . 10 |- ( x u. { x } ) = suc x
30	29	a1i 11	. . . . . . . . 9 |- ( X e. A -> ( x u. { x } ) = suc x )
31	27, 30	breq12d 4408	. . . . . . . 8 |- ( X e. A -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
32	31	3ad2ant3 1053	. . . . . . 7 |- ( ( M e. om /\ A ~~ suc M /\ X e. A ) -> ( ( ( A \ { X } ) u. { X } ) ~~ ( x u. { x } ) <-> A ~~ suc x ) )
33	32	adantr 472	. . . . . 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 7636	. . . . . . . . . . 11 |- ( A ~~ suc M -> suc M ~~ A )
35		entr 7639	. . . . . . . . . . . . 13 |- ( ( suc M ~~ A /\ A ~~ suc x ) -> suc M ~~ suc x )
36		peano2 6732	. . . . . . . . . . . . . 14 |- ( x e. om -> suc x e. om )
37		nneneq 7773	. . . . . . . . . . . . . 14 |- ( ( suc M e. om /\ suc x e. om ) -> ( suc M ~~ suc x <-> suc M = suc x ) )
38	36, 37	sylan2 482	. . . . . . . . . . . . 13 |- ( ( suc M e. om /\ x e. om ) -> ( suc M ~~ suc x <-> suc M = suc x ) )
39	35, 38	syl5ib 227	. . . . . . . . . . . 12 |- ( ( suc M e. om /\ x e. om ) -> ( ( suc M ~~ A /\ A ~~ suc x ) -> suc M = suc x ) )
40	39	expd 443	. . . . . . . . . . 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 479	. . . . . . . . 9 |- ( ( M e. om /\ x e. om ) -> ( A ~~ suc M -> ( A ~~ suc x -> suc M = suc x ) ) )
43	42	imp 436	. . . . . . . 8 |- ( ( ( M e. om /\ x e. om ) /\ A ~~ suc M ) -> ( A ~~ suc x -> suc M = suc x ) )
44	43	an32s 821	. . . . . . 7 |- ( ( ( M e. om /\ A ~~ suc M ) /\ x e. om ) -> ( A ~~ suc x -> suc M = suc x ) )
45	44	3adantl3 1188	. . . . . 6 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( A ~~ suc x -> suc M = suc x ) )
46	33, 45	sylbid 223	. . . . 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 6734	. . . . . . 7 |- ( ( M e. om /\ x e. om ) -> ( suc M = suc x <-> M = x ) )
48	47	biimpd 212	. . . . . 6 |- ( ( M e. om /\ x e. om ) -> ( suc M = suc x -> M = x ) )
49	48	3ad2antl1 1192	. . . . 5 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( suc M = suc x -> M = x ) )
50	26, 46, 49	3syld 56	. . . 4 |- ( ( ( M e. om /\ A ~~ suc M /\ X e. A ) /\ x e. om ) -> ( ( A \ { X } ) ~~ x -> M = x ) )
51		breq2 4399	. . . . 5 |- ( M = x -> ( ( A \ { X } ) ~~ M <-> ( A \ { X } ) ~~ x ) )
52	51	biimprcd 233	. . . 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 2871	. 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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   E.wrex 2757   _Vcvv 3031   \ cdif 3387   u. cun 3388   i^i cin 3389   (/)c0 3722   {csn 3959   class class class wbr 4395   Ord word 5429   suc csuc 5432   omcom 6711   ~~ cen 7584   Fincfn 7587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-er 7381  df-en 7588  df-fin 7591

This theorem is referenced by:  enp1i  7824  findcard  7828  findcard2  7829  en2eleq  8457  en2other2  8458  mreexexlem4d  15631  f1otrspeq  17166  pmtrf  17174  pmtrmvd  17175  pmtrfinv  17180