Theorem limensuci 4653

Description: A limit ordinal is equinumerous to its successor.

Hypothesis

Ref	Expression
limensuci.1	|- Lim A

Assertion

Ref	Expression
limensuci	|- (A e. B -> A ~~ suc A)

Proof of Theorem limensuci

Step	Hyp	Ref	Expression
1		incom 2260	. . . . 5 |- ((A \ {(/)}) i^i {(/)}) = ({(/)} i^i (A \ {(/)}))
2		difdisj 2390	. . . . 5 |- ({(/)} i^i (A \ {(/)})) = (/)
3	1, 2	eqtri 1538	. . . 4 |- ((A \ {(/)}) i^i {(/)}) = (/)
4		limensuci.1	. . . . . . 7 |- Lim A
5		limord 3032	. . . . . . 7 |- (Lim A -> Ord A)
6	4, 5	ax-mp 7	. . . . . 6 |- Ord A
7		ordirr 2993	. . . . . 6 |- (Ord A -> -. A e. A)
8	6, 7	ax-mp 7	. . . . 5 |- -. A e. A
9		disjsn 2502	. . . . 5 |- ((A i^i {A}) = (/) <-> -. A e. A)
10	8, 9	mpbir 188	. . . 4 |- (A i^i {A}) = (/)
11		unen 4575	. . . 4 |- ((((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) /\ (((A \ {(/)}) i^i {(/)}) = (/) /\ (A i^i {A}) = (/))) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
12	3, 10, 11	mpanr12 715	. . 3 |- (((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
13		ensymg 4552	. . . 4 |- ((A \ {(/)}) e. V -> (A ~~ (A \ {(/)}) -> (A \ {(/)}) ~~ A))
14		difexg 2796	. . . 4 |- (A e. B -> (A \ {(/)}) e. V)
15	4	limenpsi 4652	. . . 4 |- (A e. B -> A ~~ (A \ {(/)}))
16	13, 14, 15	sylc 68	. . 3 |- (A e. B -> (A \ {(/)}) ~~ A)
17		0ex 2785	. . . 4 |- (/) e. V
18		en2sn 4572	. . . 4 |- (((/) e. V /\ A e. B) -> {(/)} ~~ {A})
19	17, 18	mpan 699	. . 3 |- (A e. B -> {(/)} ~~ {A})
20	12, 16, 19	sylanc 473	. 2 |- (A e. B -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
21		0ellim 3035	. . . . . 6 |- (Lim A -> (/) e. A)
22	4, 21	ax-mp 7	. . . . 5 |- (/) e. A
23	17	snss 2525	. . . . 5 |- ((/) e. A <-> {(/)} (_ A)
24	22, 23	mpbi 187	. . . 4 |- {(/)} (_ A
25		undif 2397	. . . 4 |- ({(/)} (_ A <-> ({(/)} u. (A \ {(/)})) = A)
26	24, 25	mpbi 187	. . 3 |- ({(/)} u. (A \ {(/)})) = A
27		uncom 2228	. . 3 |- ({(/)} u. (A \ {(/)})) = ((A \ {(/)}) u. {(/)})
28	26, 27	eqtr3i 1540	. 2 |- A = ((A \ {(/)}) u. {(/)})
29		df-suc 2981	. 2 |- suc A = (A u. {A})
30	20, 28, 29	3brtr4g 2720	1 |- (A e. B -> A ~~ suc A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221   = wceq 992   e. wcel 994  Vcvv 1857   \ cdif 2096   u. cun 2097   i^i cin 2098   (_ wss 2099  (/)c0 2332  {csn 2467   class class class wbr 2692  Ord word 2974  Lim wlim 2976  suc csuc 2977   ~~ cen 4505

This theorem is referenced by:  limensuc 4654  omensuc 4783

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-1o 4269  df-er 4401  df-en 4509  df-dom 4510

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