Theorem limensuci 3401

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 1636	. . . . . 6 |- ((A \ {(/)}) i^i {(/)}) = ({(/)} i^i (A \ {(/)}))
2		difdisj 1758	. . . . . 6 |- ({(/)} i^i (A \ {(/)})) = (/)
3	1, 2	eqtr 1119	. . . . 5 |- ((A \ {(/)}) i^i {(/)}) = (/)
4		limensuci.1	. . . . . . . 8 |- Lim A
5		limord 2283	. . . . . . . 8 |- (Lim A -> Ord A)
6	4, 5	ax-mp 6	. . . . . . 7 |- Ord A
7		ordeirr 2217	. . . . . . 7 |- (Ord A -> -. A e. A)
8	6, 7	ax-mp 6	. . . . . 6 |- -. A e. A
9		disjsn 1836	. . . . . 6 |- ((A i^i {A}) = (/) <-> -. A e. A)
10	8, 9	mpbir 165	. . . . 5 |- (A i^i {A}) = (/)
11	3, 10	pm3.2i 234	. . . 4 |- (((A \ {(/)}) i^i {(/)}) = (/) /\ (A i^i {A}) = (/))
12		unen 3338	. . . 4 |- ((((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) /\ (((A \ {(/)}) i^i {(/)}) = (/) /\ (A i^i {A}) = (/))) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
13	11, 12	mpan2 519	. . 3 |- (((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
14		ensymg 3316	. . . 4 |- ((A \ {(/)}) e. V -> (A ~~ (A \ {(/)}) -> (A \ {(/)}) ~~ A))
15		difexg 1703	. . . 4 |- (A e. B -> (A \ {(/)}) e. V)
16	4	limenpsi 3400	. . . 4 |- (A e. B -> A ~~ (A \ {(/)}))
17	14, 15, 16	sylc 62	. . 3 |- (A e. B -> (A \ {(/)}) ~~ A)
18		0ex 1745	. . . 4 |- (/) e. V
19		en2sn 3336	. . . 4 |- (((/) e. V /\ A e. B) -> {(/)} ~~ {A})
20	18, 19	mpan 518	. . 3 |- (A e. B -> {(/)} ~~ {A})
21	13, 17, 20	sylanc 361	. 2 |- (A e. B -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
22		0ellim 2285	. . . . . 6 |- (Lim A -> (/) e. A)
23	4, 22	ax-mp 6	. . . . 5 |- (/) e. A
24	18	snss 1849	. . . . 5 |- ((/) e. A <-> {(/)} (_ A)
25	23, 24	mpbi 164	. . . 4 |- {(/)} (_ A
26		ssundif 1764	. . . 4 |- ({(/)} (_ A <-> ({(/)} u. (A \ {(/)})) = A)
27	25, 26	mpbi 164	. . 3 |- ({(/)} u. (A \ {(/)})) = A
28		uncom 1604	. . 3 |- ({(/)} u. (A \ {(/)})) = ((A \ {(/)}) u. {(/)})
29	27, 28	eqtr3 1121	. 2 |- A = ((A \ {(/)}) u. {(/)})
30		df-suc 2205	. 2 |- suc A = (A u. {A})
31	21, 29, 30	3brtr4g 2088	1 |- (A e. B -> A ~~ suc A)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   \ cdif 1484   u. cun 1485   i^i cin 1486   (_ wss 1487  (/)c0 1707  {csn 1808   class class class wbr 2054  Ord word 2198  Lim wlim 2200  suc csuc 2201   ~~ cen 3271

This theorem is referenced by:  limensuc 3402  omensuc 3483

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-1o 3104  df-er 3200  df-en 3274  df-dom 3275

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