Theorem suceloni 3170

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		ordon 3141	. . . 4 |- Ord On
2		trssord 2992	. . . . 5 |- ((Tr suc A /\ suc A (_ On /\ Ord On) -> Ord suc A)
3	2	3exp 838	. . . 4 |- (Tr suc A -> (suc A (_ On -> (Ord On -> Ord suc A)))
4	1, 3	mpii 45	. . 3 |- (Tr suc A -> (suc A (_ On -> Ord suc A))
5		onelss 3017	. . . . . . . 8 |- (A e. On -> (x e. A -> x (_ A))
6		elsn 2479	. . . . . . . . . 10 |- (x e. {A} <-> x = A)
7		eqimss 2161	. . . . . . . . . 10 |- (x = A -> x (_ A)
8	6, 7	sylbi 197	. . . . . . . . 9 |- (x e. {A} -> x (_ A)
9	8	a1i 8	. . . . . . . 8 |- (A e. On -> (x e. {A} -> x (_ A))
10	5, 9	orim12d 568	. . . . . . 7 |- (A e. On -> ((x e. A \/ x e. {A}) -> (x (_ A \/ x (_ A)))
11		df-suc 2981	. . . . . . . . 9 |- suc A = (A u. {A})
12	11	eleq2i 1581	. . . . . . . 8 |- (x e. suc A <-> x e. (A u. {A}))
13		elun 2225	. . . . . . . 8 |- (x e. (A u. {A}) <-> (x e. A \/ x e. {A}))
14	12, 13	bitr2i 172	. . . . . . 7 |- ((x e. A \/ x e. {A}) <-> x e. suc A)
15		oridm 241	. . . . . . 7 |- ((x (_ A \/ x (_ A) <-> x (_ A)
16	10, 14, 15	3imtr3g 555	. . . . . 6 |- (A e. On -> (x e. suc A -> x (_ A))
17		sssucid 3050	. . . . . . 7 |- A (_ suc A
18		sstr2 2123	. . . . . . 7 |- (x (_ A -> (A (_ suc A -> x (_ suc A))
19	17, 18	mpi 44	. . . . . 6 |- (x (_ A -> x (_ suc A)
20	16, 19	syl6 22	. . . . 5 |- (A e. On -> (x e. suc A -> x (_ suc A))
21	20	r19.21aiv 1759	. . . 4 |- (A e. On -> A.x e. suc Ax (_ suc A)
22		dftr3 2758	. . . 4 |- (Tr suc A <-> A.x e. suc Ax (_ suc A)
23	21, 22	sylibr 198	. . 3 |- (A e. On -> Tr suc A)
24		onss 3146	. . . . . 6 |- (A e. On -> A (_ On)
25		snssi 2530	. . . . . 6 |- (A e. On -> {A} (_ On)
26	24, 25	jca 286	. . . . 5 |- (A e. On -> (A (_ On /\ {A} (_ On))
27		unss 2256	. . . . 5 |- ((A (_ On /\ {A} (_ On) <-> (A u. {A}) (_ On)
28	26, 27	sylib 196	. . . 4 |- (A e. On -> (A u. {A}) (_ On)
29	28, 11	syl5ss 2157	. . 3 |- (A e. On -> suc A (_ On)
30	4, 23, 29	sylc 68	. 2 |- (A e. On -> Ord suc A)
31		sucexg 3167	. . 3 |- (A e. On -> suc A e. V)
32		elong 2983	. . 3 |- (suc A e. V -> (suc A e. On <-> Ord suc A))
33	31, 32	syl 10	. 2 |- (A e. On -> (suc A e. On <-> Ord suc A))
34	30, 33	mpbird 194	1 |- (A e. On -> suc A e. On)

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994  A.wral 1691  Vcvv 1857   u. cun 2097   (_ wss 2099  {csn 2467  Tr wtr 2754  Ord word 2974  Oncon0 2975  suc csuc 2977

This theorem is referenced by:  ordsuc 3171  unon 3185  onsuci 3192  ordunisuc2 3198  ordzsl 3199  tfindsg 3213  dfom2 3220  findsg 3245  tfrlem12 4223  oasuc 4299  omsuc 4301  oesuc 4302  oacl 4306  oneo 4348  oelim2 4358  nnacom 4373  nneob 4395  r1ord 4801  rankwflem 4811  rankr1 4820  bndrank 4828  r1pw 4832  cardsucinf 4991  omsublim 11448

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-suc 2981

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