Theorem dif1enOLD 10173

Description: If a set A is equinumerous to the successor of natural number M, then A with an element removed is equinumerous to the predecessor of M. (Contributed by Jeff Madsen 2-Sep-2009.)

Hypothesis

Ref	Expression
dif1enOLD.1	|- A e. _V

Assertion

Ref	Expression
dif1enOLD	|- ((M e. om /\ A ~~ suc M /\ X e. A) -> (A \ {X}) ~~ M)

Proof of Theorem dif1enOLD

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . 6 |- (x = X -> (x e. A <-> X e. A))
2		sneq 3054	. . . . . . . . 9 |- (x = X -> {x} = {X})
3	2	difeq2d 2726	. . . . . . . 8 |- (x = X -> (A \ {x}) = (A \ {X}))
4	3	breq1d 3348	. . . . . . 7 |- (x = X -> ((A \ {x}) ~~ M <-> (A \ {X}) ~~ M))
5	4	imbi2d 674	. . . . . 6 |- (x = X -> (((M e. om /\ A ~~ suc M) -> (A \ {x}) ~~ M) <-> ((M e. om /\ A ~~ suc M) -> (A \ {X}) ~~ M)))
6	1, 5	imbi12d 688	. . . . 5 |- (x = X -> ((x e. A -> ((M e. om /\ A ~~ suc M) -> (A \ {x}) ~~ M)) <-> (X e. A -> ((M e. om /\ A ~~ suc M) -> (A \ {X}) ~~ M))))
7		snssi 3129	. . . . . . . . 9 |- (x e. A -> {x} C_ A)
8		undif 2954	. . . . . . . . . . 11 |- ({x} C_ A <-> ({x} u. (A \ {x})) = A)
9	8	biimpi 168	. . . . . . . . . 10 |- ({x} C_ A -> ({x} u. (A \ {x})) = A)
10		uncom 2744	. . . . . . . . . 10 |- ({x} u. (A \ {x})) = ((A \ {x}) u. {x})
11	9, 10	syl5reqr 1943	. . . . . . . . 9 |- ({x} C_ A -> A = ((A \ {x}) u. {x}))
12		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
13	12	snid 3069	. . . . . . . . . . . 12 |- x e. {x}
14		elndif 2732	. . . . . . . . . . . 12 |- (x e. {x} -> -. x e. (A \ {x}))
15	13, 14	ax-mp 7	. . . . . . . . . . 11 |- -. x e. (A \ {x})
16		dif1enOLD.1	. . . . . . . . . . . . 13 |- A e. _V
17		difexg 3458	. . . . . . . . . . . . 13 |- (A e. _V -> (A \ {x}) e. _V)
18	16, 17	ax-mp 7	. . . . . . . . . . . 12 |- (A \ {x}) e. _V
19	18, 12	unsnen 5985	. . . . . . . . . . 11 |- (-. x e. (A \ {x}) -> ((A \ {x}) u. {x}) ~~ suc (card` (A \ {x})))
20	15, 19	ax-mp 7	. . . . . . . . . 10 |- ((A \ {x}) u. {x}) ~~ suc (card` (A \ {x}))
21		breq1 3341	. . . . . . . . . 10 |- (A = ((A \ {x}) u. {x}) -> (A ~~ suc (card` (A \ {x})) <-> ((A \ {x}) u. {x}) ~~ suc (card` (A \ {x}))))
22	20, 21	mpbiri 211	. . . . . . . . 9 |- (A = ((A \ {x}) u. {x}) -> A ~~ suc (card` (A \ {x})))
23	7, 11, 22	3syl 24	. . . . . . . 8 |- (x e. A -> A ~~ suc (card` (A \ {x})))
24		onomeneq 5612	. . . . . . . . . . . . . 14 |- ((suc (card` (A \ {x})) e. On /\ suc M e. om) -> (suc (card` (A \ {x})) ~~ suc M <-> suc (card` (A \ {x})) = suc M))
25		cardon 5976	. . . . . . . . . . . . . . 15 |- (card` (A \ {x})) e. On
26	25	onsuci 3919	. . . . . . . . . . . . . 14 |- suc (card` (A \ {x})) e. On
27		peano2 3972	. . . . . . . . . . . . . 14 |- (M e. om -> suc M e. om)
28	24, 26, 27	sylancr 526	. . . . . . . . . . . . 13 |- (M e. om -> (suc (card` (A \ {x})) ~~ suc M <-> suc (card` (A \ {x})) = suc M))
29		suc11 3773	. . . . . . . . . . . . . 14 |- (((card` (A \ {x})) e. On /\ M e. On) -> (suc (card` (A \ {x})) = suc M <-> (card` (A \ {x})) = M))
30		nnon 3957	. . . . . . . . . . . . . 14 |- (M e. om -> M e. On)
31	29, 25, 30	sylancr 526	. . . . . . . . . . . . 13 |- (M e. om -> (suc (card` (A \ {x})) = suc M <-> (card` (A \ {x})) = M))
32	28, 31	bitrd 587	. . . . . . . . . . . 12 |- (M e. om -> (suc (card` (A \ {x})) ~~ suc M <-> (card` (A \ {x})) = M))
33		cardid 5977	. . . . . . . . . . . . . 14 |- (card` (A \ {x})) ~~ (A \ {x})
34	18, 33	ensymi 5472	. . . . . . . . . . . . 13 |- (A \ {x}) ~~ (card` (A \ {x}))
35		breq2 3342	. . . . . . . . . . . . 13 |- ((card` (A \ {x})) = M -> ((A \ {x}) ~~ (card` (A \ {x})) <-> (A \ {x}) ~~ M))
36	34, 35	mpbii 210	. . . . . . . . . . . 12 |- ((card` (A \ {x})) = M -> (A \ {x}) ~~ M)
37	32, 36	syl6bi 231	. . . . . . . . . . 11 |- (M e. om -> (suc (card` (A \ {x})) ~~ suc M -> (A \ {x}) ~~ M))
38		entr 5473	. . . . . . . . . . 11 |- ((suc (card` (A \ {x})) ~~ A /\ A ~~ suc M) -> suc (card` (A \ {x})) ~~ suc M)
39	37, 38	syl5com 63	. . . . . . . . . 10 |- ((suc (card` (A \ {x})) ~~ A /\ A ~~ suc M) -> (M e. om -> (A \ {x}) ~~ M))
40		fvex 4689	. . . . . . . . . . . 12 |- (card` (A \ {x})) e. _V
41	40	sucex 3892	. . . . . . . . . . 11 |- suc (card` (A \ {x})) e. _V
42	41	ensym 5471	. . . . . . . . . 10 |- (A ~~ suc (card` (A \ {x})) -> suc (card` (A \ {x})) ~~ A)
43	39, 42	sylan 497	. . . . . . . . 9 |- ((A ~~ suc (card` (A \ {x})) /\ A ~~ suc M) -> (M e. om -> (A \ {x}) ~~ M))
44	43	ex 402	. . . . . . . 8 |- (A ~~ suc (card` (A \ {x})) -> (A ~~ suc M -> (M e. om -> (A \ {x}) ~~ M)))
45	23, 44	syl 12	. . . . . . 7 |- (x e. A -> (A ~~ suc M -> (M e. om -> (A \ {x}) ~~ M)))
46	45	com23 36	. . . . . 6 |- (x e. A -> (M e. om -> (A ~~ suc M -> (A \ {x}) ~~ M)))
47	46	imp3a 388	. . . . 5 |- (x e. A -> ((M e. om /\ A ~~ suc M) -> (A \ {x}) ~~ M))
48	6, 47	vtoclg 2346	. . . 4 |- (X e. A -> (X e. A -> ((M e. om /\ A ~~ suc M) -> (A \ {X}) ~~ M)))
49	48	pm2.43i 78	. . 3 |- (X e. A -> ((M e. om /\ A ~~ suc M) -> (A \ {X}) ~~ M))
50	49	com12 14	. 2 |- ((M e. om /\ A ~~ suc M) -> (X e. A -> (A \ {X}) ~~ M))
51	50	3impia 1064	1 |- ((M e. om /\ A ~~ suc M /\ X e. A) -> (A \ {X}) ~~ M)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292   \ cdif 2590   u. cun 2591   C_ wss 2593  {csn 3044   class class class wbr 3338  Oncon0 3657  suc csuc 3659  omcom 3949  ` cfv 3998   ~~ cen 5423  cardccrd 5859

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  ax-ac 5906

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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-int 3215  df-iun 3257  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-dom 5428  df-sdom 5429  df-card 5862

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