Theorem suceloni 3894

Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41.

Assertion

Ref	Expression
suceloni	|- (A e. On -> suc A e. On)

Proof of Theorem suceloni

Step	Hyp	Ref	Expression
1		onelss 3705	. . . . . . . 8 |- (A e. On -> (x e. A -> x C_ A))
2		elsn 3058	. . . . . . . . . 10 |- (x e. {A} <-> x = A)
3		eqimss 2665	. . . . . . . . . 10 |- (x = A -> x C_ A)
4	2, 3	sylbi 216	. . . . . . . . 9 |- (x e. {A} -> x C_ A)
5	4	a1i 8	. . . . . . . 8 |- (A e. On -> (x e. {A} -> x C_ A))
6	1, 5	orim12d 624	. . . . . . 7 |- (A e. On -> ((x e. A \/ x e. {A}) -> (x C_ A \/ x C_ A)))
7		df-suc 3663	. . . . . . . . 9 |- suc A = (A u. {A})
8	7	eleq2i 1961	. . . . . . . 8 |- (x e. suc A <-> x e. (A u. {A}))
9		elun 2741	. . . . . . . 8 |- (x e. (A u. {A}) <-> (x e. A \/ x e. {A}))
10	8, 9	bitr2i 191	. . . . . . 7 |- ((x e. A \/ x e. {A}) <-> x e. suc A)
11		oridm 262	. . . . . . 7 |- ((x C_ A \/ x C_ A) <-> x C_ A)
12	6, 10, 11	3imtr3g 611	. . . . . 6 |- (A e. On -> (x e. suc A -> x C_ A))
13		sssucid 3742	. . . . . 6 |- A C_ suc A
14		sstr2 2623	. . . . . 6 |- (x C_ A -> (A C_ suc A -> x C_ suc A))
15	12, 13, 14	syl6mpi 68	. . . . 5 |- (A e. On -> (x e. suc A -> x C_ suc A))
16	15	r19.21aiv 2175	. . . 4 |- (A e. On -> A.x e. suc Ax C_ suc A)
17		dftr3 3415	. . . 4 |- (Tr suc A <-> A.x e. suc Ax C_ suc A)
18	16, 17	sylibr 217	. . 3 |- (A e. On -> Tr suc A)
19		onss 3869	. . . . . 6 |- (A e. On -> A C_ On)
20		snssi 3129	. . . . . 6 |- (A e. On -> {A} C_ On)
21	19, 20	jca 310	. . . . 5 |- (A e. On -> (A C_ On /\ {A} C_ On))
22		unss 2780	. . . . 5 |- ((A C_ On /\ {A} C_ On) <-> (A u. {A}) C_ On)
23	21, 22	sylib 215	. . . 4 |- (A e. On -> (A u. {A}) C_ On)
24	23, 7	syl5ss 2661	. . 3 |- (A e. On -> suc A C_ On)
25		ordon 3863	. . . 4 |- Ord On
26		trssord 3675	. . . . 5 |- ((Tr suc A /\ suc A C_ On /\ Ord On) -> Ord suc A)
27	26	3exp 1066	. . . 4 |- (Tr suc A -> (suc A C_ On -> (Ord On -> Ord suc A)))
28	25, 27	mpii 56	. . 3 |- (Tr suc A -> (suc A C_ On -> Ord suc A))
29	18, 24, 28	sylc 83	. 2 |- (A e. On -> Ord suc A)
30		sucexg 3891	. . 3 |- (A e. On -> suc A e. _V)
31		elong 3665	. . 3 |- (suc A e. _V -> (suc A e. On <-> Ord suc A))
32	30, 31	syl 12	. 2 |- (A e. On -> (suc A e. On <-> Ord suc A))
33	29, 32	mpbird 213	1 |- (A e. On -> suc A e. On)

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   u. cun 2591   C_ wss 2593  {csn 3044  Tr wtr 3411  Ord word 3656  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  ordsuc 3895  unon 3910  onsuci 3919  ordunisuc2 3926  ordzsl 3927  onzsl 3928  tfindsg 3944  dfom2 3951  findsg 3980  tfrlem12 5130  oasuc 5208  omsuc 5210  oesuc 5211  oacl 5215  oneo 5260  oelim2 5270  nnacom 5288  nneob 5312  r1ord 5766  rankwflem 5776  rankr1 5785  bndrank 5793  r1pw 5797  omsublim 5887  cardsucinf 5993  cartarlim 15282  omsublimOLD 15396

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-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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  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-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663

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