Theorem dif1en 10172

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.)

Hypothesis

Ref	Expression
dif1en.1	|- A e. _V

Assertion

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

Proof of Theorem dif1en

Step	Hyp	Ref	Expression
1		breq2 3342	. . . . . . 7 |- (x = suc M -> (A ~~ x <-> A ~~ suc M))
2	1	rcla4ev 2381	. . . . . 6 |- ((suc M e. om /\ A ~~ suc M) -> E.x e. om A ~~ x)
3		isfi 5441	. . . . . 6 |- (A e. Fin <-> E.x e. om A ~~ x)
4	2, 3	sylibr 217	. . . . 5 |- ((suc M e. om /\ A ~~ suc M) -> A e. Fin)
5		peano2 3972	. . . . 5 |- (M e. om -> suc M e. om)
6	4, 5	sylan 497	. . . 4 |- ((M e. om /\ A ~~ suc M) -> A e. Fin)
7		finfin 10167	. . . . 5 |- (A e. Fin -> (A \ {X}) e. Fin)
8		isfi 5441	. . . . 5 |- ((A \ {X}) e. Fin <-> E.x e. om (A \ {X}) ~~ x)
9	7, 8	sylib 215	. . . 4 |- (A e. Fin -> E.x e. om (A \ {X}) ~~ x)
10	6, 9	syl 12	. . 3 |- ((M e. om /\ A ~~ suc M) -> E.x e. om (A \ {X}) ~~ x)
11	10	3adant3 896	. 2 |- ((M e. om /\ A ~~ suc M /\ X e. A) -> E.x e. om (A \ {X}) ~~ x)
12		incom 2787	. . . . . . . . . 10 |- ((A \ {X}) i^i {X}) = ({X} i^i (A \ {X}))
13		difdisj 2945	. . . . . . . . . 10 |- ({X} i^i (A \ {X})) = (/)
14	12, 13	eqtri 1908	. . . . . . . . 9 |- ((A \ {X}) i^i {X}) = (/)
15		unen 5493	. . . . . . . . . 10 |- ((((A \ {X}) ~~ x /\ {X} ~~ {x}) /\ (((A \ {X}) i^i {X}) = (/) /\ (x i^i {x}) = (/))) -> ((A \ {X}) u. {X}) ~~ (x u. {x}))
16	15	an4s 566	. . . . . . . . 9 |- ((((A \ {X}) ~~ x /\ ((A \ {X}) i^i {X}) = (/)) /\ ({X} ~~ {x} /\ (x i^i {x}) = (/))) -> ((A \ {X}) u. {X}) ~~ (x u. {x}))
17	14, 16	mpanl2 771	. . . . . . . 8 |- (((A \ {X}) ~~ x /\ ({X} ~~ {x} /\ (x i^i {x}) = (/))) -> ((A \ {X}) u. {X}) ~~ (x u. {x}))
18	17	expcom 403	. . . . . . 7 |- (({X} ~~ {x} /\ (x i^i {x}) = (/)) -> ((A \ {X}) ~~ x -> ((A \ {X}) u. {X}) ~~ (x u. {x})))
19		visset 2295	. . . . . . . 8 |- x e. _V
20		en2sn 5490	. . . . . . . 8 |- ((X e. A /\ x e. _V) -> {X} ~~ {x})
21	19, 20	mpan2 760	. . . . . . 7 |- (X e. A -> {X} ~~ {x})
22		nnord 3959	. . . . . . . 8 |- (x e. om -> Ord x)
23		orddisj 3701	. . . . . . . 8 |- (Ord x -> (x i^i {x}) = (/))
24	22, 23	syl 12	. . . . . . 7 |- (x e. om -> (x i^i {x}) = (/))
25	18, 21, 24	syl2an 503	. . . . . 6 |- ((X e. A /\ x e. om) -> ((A \ {X}) ~~ x -> ((A \ {X}) u. {X}) ~~ (x u. {x})))
26	25	3ad2antl3 1040	. . . . 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 3131	. . . . . . . . 9 |- (X e. A -> ((A \ {X}) u. {X}) = A)
28		df-suc 3663	. . . . . . . . . . 11 |- suc x = (x u. {x})
29	28	eqcomi 1888	. . . . . . . . . 10 |- (x u. {x}) = suc x
30	29	a1i 8	. . . . . . . . 9 |- (X e. A -> (x u. {x}) = suc x)
31	27, 30	breq12d 3351	. . . . . . . 8 |- (X e. A -> (((A \ {X}) u. {X}) ~~ (x u. {x}) <-> A ~~ suc x))
32	31	3ad2ant3 899	. . . . . . 7 |- ((M e. om /\ A ~~ suc M /\ X e. A) -> (((A \ {X}) u. {X}) ~~ (x u. {x}) <-> A ~~ suc x))
33	32	adantr 425	. . . . . 6 |- (((M e. om /\ A ~~ suc M /\ X e. A) /\ x e. om) -> (((A \ {X}) u. {X}) ~~ (x u. {x}) <-> A ~~ suc x))
34		ensymg 5470	. . . . . . . . . . . 12 |- (suc M e. om -> (A ~~ suc M -> suc M ~~ A))
35	34	adantr 425	. . . . . . . . . . 11 |- ((suc M e. om /\ x e. om) -> (A ~~ suc M -> suc M ~~ A))
36		nneneq 5606	. . . . . . . . . . . . . 14 |- ((suc M e. om /\ suc x e. om) -> (suc M ~~ suc x <-> suc M = suc x))
37		peano2 3972	. . . . . . . . . . . . . 14 |- (x e. om -> suc x e. om)
38	36, 37	sylan2 500	. . . . . . . . . . . . 13 |- ((suc M e. om /\ x e. om) -> (suc M ~~ suc x <-> suc M = suc x))
39		entr 5473	. . . . . . . . . . . . 13 |- ((suc M ~~ A /\ A ~~ suc x) -> suc M ~~ suc x)
40	38, 39	syl5bi 225	. . . . . . . . . . . 12 |- ((suc M e. om /\ x e. om) -> ((suc M ~~ A /\ A ~~ suc x) -> suc M = suc x))
41	40	exp3a 405	. . . . . . . . . . 11 |- ((suc M e. om /\ x e. om) -> (suc M ~~ A -> (A ~~ suc x -> suc M = suc x)))
42	35, 41	syld 30	. . . . . . . . . 10 |- ((suc M e. om /\ x e. om) -> (A ~~ suc M -> (A ~~ suc x -> suc M = suc x)))
43	42, 5	sylan 497	. . . . . . . . 9 |- ((M e. om /\ x e. om) -> (A ~~ suc M -> (A ~~ suc x -> suc M = suc x)))
44	43	imp 377	. . . . . . . 8 |- (((M e. om /\ x e. om) /\ A ~~ suc M) -> (A ~~ suc x -> suc M = suc x))
45	44	an1rs 547	. . . . . . 7 |- (((M e. om /\ A ~~ suc M) /\ x e. om) -> (A ~~ suc x -> suc M = suc x))
46	45	3adantl3 1034	. . . . . 6 |- (((M e. om /\ A ~~ suc M /\ X e. A) /\ x e. om) -> (A ~~ suc x -> suc M = suc x))
47	33, 46	sylbid 220	. . . . 5 |- (((M e. om /\ A ~~ suc M /\ X e. A) /\ x e. om) -> (((A \ {X}) u. {X}) ~~ (x u. {x}) -> suc M = suc x))
48		peano4 3974	. . . . . . 7 |- ((M e. om /\ x e. om) -> (suc M = suc x <-> M = x))
49	48	biimpd 170	. . . . . 6 |- ((M e. om /\ x e. om) -> (suc M = suc x -> M = x))
50	49	3ad2antl1 1038	. . . . 5 |- (((M e. om /\ A ~~ suc M /\ X e. A) /\ x e. om) -> (suc M = suc x -> M = x))
51	26, 47, 50	3syld 31	. . . 4 |- (((M e. om /\ A ~~ suc M /\ X e. A) /\ x e. om) -> ((A \ {X}) ~~ x -> M = x))
52		breq2 3342	. . . . 5 |- (M = x -> ((A \ {X}) ~~ M <-> (A \ {X}) ~~ x))
53	52	biimprcd 173	. . . 4 |- ((A \ {X}) ~~ x -> (M = x -> (A \ {X}) ~~ M))
54	51, 53	sylcom 62	. . 3 |- (((M e. om /\ A ~~ suc M /\ X e. A) /\ x e. om) -> ((A \ {X}) ~~ x -> (A \ {X}) ~~ M))
55	54	r19.23adva 2216	. 2 |- ((M e. om /\ A ~~ suc M /\ X e. A) -> (E.x e. om (A \ {X}) ~~ x -> (A \ {X}) ~~ M))
56	11, 55	mpd 29	1 |- ((M e. om /\ A ~~ suc M /\ X e. A) -> (A \ {X}) ~~ M)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044   class class class wbr 3338  Ord word 3656  suc csuc 3659  omcom 3949   ~~ cen 5423  Fincfn 5426

This theorem is referenced by:  indexfi 10174  dif1card 10177  findcard 10178  findcard2 15745  indexfiOLD 15755

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1o 5177  df-er 5318  df-en 5427  df-fin 5430

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