Theorem limensuci 5600

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		difexg 3458	. . . 4 |- (A e. B -> (A \ {(/)}) e. _V)
2		limensuci.1	. . . . 5 |- Lim A
3	2	limenpsi 5599	. . . 4 |- (A e. B -> A ~~ (A \ {(/)}))
4		ensymg 5470	. . . 4 |- ((A \ {(/)}) e. _V -> (A ~~ (A \ {(/)}) -> (A \ {(/)}) ~~ A))
5	1, 3, 4	sylc 83	. . 3 |- (A e. B -> (A \ {(/)}) ~~ A)
6		0ex 3446	. . . 4 |- (/) e. _V
7		en2sn 5490	. . . 4 |- (((/) e. _V /\ A e. B) -> {(/)} ~~ {A})
8	6, 7	mpan 759	. . 3 |- (A e. B -> {(/)} ~~ {A})
9		incom 2787	. . . . 5 |- ((A \ {(/)}) i^i {(/)}) = ({(/)} i^i (A \ {(/)}))
10		difdisj 2945	. . . . 5 |- ({(/)} i^i (A \ {(/)})) = (/)
11	9, 10	eqtri 1908	. . . 4 |- ((A \ {(/)}) i^i {(/)}) = (/)
12		limord 3723	. . . . . . 7 |- (Lim A -> Ord A)
13	2, 12	ax-mp 7	. . . . . 6 |- Ord A
14		ordirr 3676	. . . . . 6 |- (Ord A -> -. A e. A)
15	13, 14	ax-mp 7	. . . . 5 |- -. A e. A
16		disjsn 3089	. . . . 5 |- ((A i^i {A}) = (/) <-> -. A e. A)
17	15, 16	mpbir 207	. . . 4 |- (A i^i {A}) = (/)
18		unen 5493	. . . 4 |- ((((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) /\ (((A \ {(/)}) i^i {(/)}) = (/) /\ (A i^i {A}) = (/))) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
19	11, 17, 18	mpanr12 778	. . 3 |- (((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
20	5, 8, 19	syl11anc 524	. 2 |- (A e. B -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
21		0ellim 3726	. . . . . 6 |- (Lim A -> (/) e. A)
22	2, 21	ax-mp 7	. . . . 5 |- (/) e. A
23	6	snss 3122	. . . . 5 |- ((/) e. A <-> {(/)} C_ A)
24	22, 23	mpbi 206	. . . 4 |- {(/)} C_ A
25		undif 2954	. . . 4 |- ({(/)} C_ A <-> ({(/)} u. (A \ {(/)})) = A)
26	24, 25	mpbi 206	. . 3 |- ({(/)} u. (A \ {(/)})) = A
27		uncom 2744	. . 3 |- ({(/)} u. (A \ {(/)})) = ((A \ {(/)}) u. {(/)})
28	26, 27	eqtr3i 1910	. 2 |- A = ((A \ {(/)}) u. {(/)})
29		df-suc 3663	. 2 |- suc A = (A u. {A})
30	20, 28, 29	3brtr4g 3369	1 |- (A e. B -> A ~~ suc A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338  Ord word 3656  Lim wlim 3658  suc csuc 3659   ~~ cen 5423

This theorem is referenced by:  limensuc 5601  omensuc 5744

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-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

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