Theorem suceloni 3062

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 2987	. . . 4 |- Ord On
2		trssord 2965	. . . . 5 |- ((Tr suc A /\ suc A (_ On /\ Ord On) -> Ord suc A)
3	2	3exp 832	. . . 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		onelsst 3000	. . . . . . . 8 |- (A e. On -> (x e. A -> x (_ A))
6		elsn 2421	. . . . . . . . . 10 |- (x e. {A} <-> x = A)
7		eqimss 2109	. . . . . . . . . 10 |- (x = A -> x (_ A)
8	6, 7	sylbi 199	. . . . . . . . 9 |- (x e. {A} -> x (_ A)
9	8	a1i 8	. . . . . . . 8 |- (A e. On -> (x e. {A} -> x (_ A))
10	5, 9	orim12d 565	. . . . . . 7 |- (A e. On -> ((x e. A \/ x e. {A}) -> (x (_ A \/ x (_ A)))
11		df-suc 2954	. . . . . . . . 9 |- suc A = (A u. {A})
12	11	eleq2i 1538	. . . . . . . 8 |- (x e. suc A <-> x e. (A u. {A}))
13		elun 2173	. . . . . . . 8 |- (x e. (A u. {A}) <-> (x e. A \/ x e. {A}))
14	12, 13	bitr2 174	. . . . . . 7 |- ((x e. A \/ x e. {A}) <-> x e. suc A)
15		oridm 243	. . . . . . 7 |- ((x (_ A \/ x (_ A) <-> x (_ A)
16	10, 14, 15	3imtr3g 552	. . . . . 6 |- (A e. On -> (x e. suc A -> x (_ A))
17		sssucid 3047	. . . . . . 7 |- A (_ suc A
18		sstr2 2071	. . . . . . 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 1713	. . . 4 |- (A e. On -> A.x e. suc Ax (_ suc A)
22		dftr3 2684	. . . 4 |- (Tr suc A <-> A.x e. suc Ax (_ suc A)
23	21, 22	sylibr 200	. . 3 |- (A e. On -> Tr suc A)
24		onsst 2992	. . . . . 6 |- (A e. On -> A (_ On)
25		snssi 2466	. . . . . 6 |- (A e. On -> {A} (_ On)
26	24, 25	jca 288	. . . . 5 |- (A e. On -> (A (_ On /\ {A} (_ On))
27		unss 2204	. . . . 5 |- ((A (_ On /\ {A} (_ On) <-> (A u. {A}) (_ On)
28	26, 27	sylib 198	. . . 4 |- (A e. On -> (A u. {A}) (_ On)
29	28, 11	syl5ss 2105	. . 3 |- (A e. On -> suc A (_ On)
30	4, 23, 29	sylc 68	. 2 |- (A e. On -> Ord suc A)
31		sucexg 3049	. . 3 |- (A e. On -> suc A e. V)
32		elong 2956	. . 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 196	1 |- (A e. On -> suc A e. On)

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   u. cun 2045   (_ wss 2047  {csn 2409  Tr wtr 2680  Ord word 2947  Oncon0 2948  suc csuc 2950

This theorem is referenced by:  ordsuc 3065  unon 3088  onsuc 3105  ordunisuc2 3115  ordzsl 3116  dfom2 3133  findsg 3157  tfindsg 3162  tfrlem12 3922  oasuc 4163  omsuc 4165  oesuc 4166  oacl 4170  oneo 4212  oelim2 4222  nnacom 4233  nneob 4255  r1ord 4655  rankwflem 4665  rankr1 4674  bndrank 4682  r1pw 4686

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954

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