Theorem limensuci 4492

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 2204	. . . . . 6 |- ((A \ {(/)}) i^i {(/)}) = ({(/)} i^i (A \ {(/)}))
2		difdisj 2333	. . . . . 6 |- ({(/)} i^i (A \ {(/)})) = (/)
3	1, 2	eqtr 1492	. . . . 5 |- ((A \ {(/)}) i^i {(/)}) = (/)
4		limensuci.1	. . . . . . . 8 |- Lim A
5		limord 3023	. . . . . . . 8 |- (Lim A -> Ord A)
6	4, 5	ax-mp 7	. . . . . . 7 |- Ord A
7		ordirr 2961	. . . . . . 7 |- (Ord A -> -. A e. A)
8	6, 7	ax-mp 7	. . . . . 6 |- -. A e. A
9		disjsn 2437	. . . . . 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 4420	. . . 4 |- ((((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) /\ (((A \ {(/)}) i^i {(/)}) = (/) /\ (A i^i {A}) = (/))) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
13	11, 12	mpan2 695	. . 3 |- (((A \ {(/)}) ~~ A /\ {(/)} ~~ {A}) -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
14		ensymg 4398	. . . 4 |- ((A \ {(/)}) e. V -> (A ~~ (A \ {(/)}) -> (A \ {(/)}) ~~ A))
15		difexg 2717	. . . 4 |- (A e. B -> (A \ {(/)}) e. V)
16	4	limenpsi 4491	. . . 4 |- (A e. B -> A ~~ (A \ {(/)}))
17	14, 15, 16	sylc 68	. . 3 |- (A e. B -> (A \ {(/)}) ~~ A)
18		0ex 2706	. . . 4 |- (/) e. V
19		en2sn 4418	. . . 4 |- (((/) e. V /\ A e. B) -> {(/)} ~~ {A})
20	18, 19	mpan 694	. . 3 |- (A e. B -> {(/)} ~~ {A})
21	13, 17, 20	sylanc 471	. 2 |- (A e. B -> ((A \ {(/)}) u. {(/)}) ~~ (A u. {A}))
22		0ellim 3026	. . . . . 6 |- (Lim A -> (/) e. A)
23	4, 22	ax-mp 7	. . . . 5 |- (/) e. A
24	18	snss 2457	. . . . 5 |- ((/) e. A <-> {(/)} (_ A)
25	23, 24	mpbi 189	. . . 4 |- {(/)} (_ A
26		undif 2339	. . . 4 |- ({(/)} (_ A <-> ({(/)} u. (A \ {(/)})) = A)
27	25, 26	mpbi 189	. . 3 |- ({(/)} u. (A \ {(/)})) = A
28		uncom 2172	. . 3 |- ({(/)} u. (A \ {(/)})) = ((A \ {(/)}) u. {(/)})
29	27, 28	eqtr3 1494	. 2 |- A = ((A \ {(/)}) u. {(/)})
30		df-suc 2949	. 2 |- suc A = (A u. {A})
31	21, 29, 30	3brtr4g 2642	1 |- (A e. B -> A ~~ suc A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807   \ cdif 2040   u. cun 2041   i^i cin 2042   (_ wss 2043  (/)c0 2276  {csn 2405   class class class wbr 2614  Ord word 2942  Lim wlim 2944  suc csuc 2945   ~~ cen 4354

This theorem is referenced by:  limensuc 4493  omensuc 4617

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-1o 4123  df-er 4251  df-en 4357  df-dom 4358

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