Theorem limensuci 4486

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 2198	. . . . . 6 |- ((A \ {(/)}) i^i {(/)}) = ({(/)} i^i (A \ {(/)}))
2		difdisj 2327	. . . . . 6 |- ({(/)} i^i (A \ {(/)})) = (/)
3	1, 2	eqtr 1487	. . . . 5 |- ((A \ {(/)}) i^i {(/)}) = (/)
4		limensuci.1	. . . . . . . 8 |- Lim A
5		limord 3018	. . . . . . . 8 |- (Lim A -> Ord A)
6	4, 5	ax-mp 7	. . . . . . 7 |- Ord A
7		ordirr 2956	. . . . . . 7 |- (Ord A -> -. A e. A)
8	6, 7	ax-mp 7	. . . . . 6 |- -. A e. A
9		disjsn 2431	. . . . . 6 |- ((A i^i {A}) = (/) <-> -. A e. A)
10	8, 9	mpbir 190	. . . . 5 |- (A i^i {A}) = (/)
11	3, 10	pm3.2i 285	. . . 4 |- (((A \ {(/)}) i^i {(/)}) = (/) /\ (A i^i {A}) = (/))
12		unen 4414	. . . 4 |- ((((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) /\ (((A \ {(/)}) i^i {(/)}) = (/) /\ (A i^i {A}) = (/))) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
13	11, 12	mpan2 694	. . 3 |- (((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
14		ensymg 4392	. . . 4 |- ((A \ {(/)}) e. V -> (A ~~ (A \ {(/)}) -> (A \ {(/)}) ~~ A))
15		difexg 2712	. . . 4 |- (A e. B -> (A \ {(/)}) e. V)
16	4	limenpsi 4485	. . . 4 |- (A e. B -> A ~~ (A \ {(/)}))
17	14, 15, 16	sylc 68	. . 3 |- (A e. B -> (A \ {(/)}) ~~ A)
18		0ex 2701	. . . 4 |- (/) e. V
19		en2sn 4412	. . . 4 |- (((/) e. V /\ A e. B) -> {(/)} ~~ {A})
20	18, 19	mpan 693	. . 3 |- (A e. B -> {(/)} ~~ {A})
21	13, 17, 20	sylanc 471	. 2 |- (A e. B -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
22		0ellim 3021	. . . . . 6 |- (Lim A -> (/) e. A)
23	4, 22	ax-mp 7	. . . . 5 |- (/) e. A
24	18	snss 2452	. . . . 5 |- ((/) e. A <-> {(/)} (_ A)
25	23, 24	mpbi 189	. . . 4 |- {(/)} (_ A
26		undif 2333	. . . 4 |- ({(/)} (_ A <-> ({(/)} u. (A \ {(/)})) = A)
27	25, 26	mpbi 189	. . 3 |- ({(/)} u. (A \ {(/)})) = A
28		uncom 2166	. . 3 |- ({(/)} u. (A \ {(/)})) = ((A \ {(/)}) u. {(/)})
29	27, 28	eqtr3 1489	. 2 |- A = ((A \ {(/)}) u. {(/)})
30		df-suc 2944	. 2 |- suc A = (A u. {A})
31	21, 29, 30	3brtr4g 2637	1 |- (A e. B -> A ~~ suc A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802   \ cdif 2034   u. cun 2035   i^i cin 2036   (_ wss 2037  (/)c0 2270  {csn 2399   class class class wbr 2609  Ord word 2937  Lim wlim 2939  suc csuc 2940   ~~ cen 4348

This theorem is referenced by:  limensuc 4487  omensuc 4609

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-1o 4117  df-er 4245  df-en 4351  df-dom 4352

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