Theorem suceloni 7592

Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)

Assertion

Ref	Expression
suceloni	⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)

Proof of Theorem suceloni

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelss 6255	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		velsn 4557	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		eqimss 3957	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
4	2, 3	sylbi 220	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ⊆ 𝐴)
5	4	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥 ⊆ 𝐴))
6	1, 5	orim12d 965	. . . . . . 7 ⊢ (𝐴 ∈ On → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴)))
7		df-suc 6219	. . . . . . . . 9 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
8	7	eleq2i 2829	. . . . . . . 8 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴}))
9		elun 4063	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10	8, 9	bitr2i 279	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11		oridm 905	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴) ↔ 𝑥 ⊆ 𝐴)
12	6, 10, 11	3imtr3g 298	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴))
13		sssucid 6290	. . . . . 6 ⊢ 𝐴 ⊆ suc 𝐴
14		sstr2 3908	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ suc 𝐴 → 𝑥 ⊆ suc 𝐴))
15	12, 13, 14	syl6mpi 67	. . . . 5 ⊢ (𝐴 ∈ On → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ suc 𝐴))
16	15	ralrimiv 3104	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17		dftr3 5165	. . . 4 ⊢ (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
18	16, 17	sylibr 237	. . 3 ⊢ (𝐴 ∈ On → Tr suc 𝐴)
19		onss 7568	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
20		snssi 4721	. . . . 5 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
21	19, 20	unssd 4100	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
22	7, 21	eqsstrid 3949	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ⊆ On)
23		ordon 7561	. . . 4 ⊢ Ord On
24		trssord 6230	. . . . 5 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25	24	3exp 1121	. . . 4 ⊢ (Tr suc 𝐴 → (suc 𝐴 ⊆ On → (Ord On → Ord suc 𝐴)))
26	23, 25	mpii 46	. . 3 ⊢ (Tr suc 𝐴 → (suc 𝐴 ⊆ On → Ord suc 𝐴))
27	18, 22, 26	sylc 65	. 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
28		sucexg 7589	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
29		elong 6221	. . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
30	28, 29	syl 17	. 2 ⊢ (𝐴 ∈ On → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
31	27, 30	mpbird 260	1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)

Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 847   = wceq 1543   ∈ wcel 2110  ∀wral 3061  Vcvv 3408   ∪ cun 3864   ⊆ wss 3866  {csn 4541  Tr wtr 5161  Ord word 6212  Oncon0 6213  suc csuc 6215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216  df-on 6217  df-suc 6219

This theorem is referenced by:  ordsuc  7593  unon  7610  onsuci  7617  ordunisuc2  7623  ordzsl  7624  onzsl  7625  tfindsg  7639  dfom2  7646  findsg  7677  tfrlem12  8125  oasuc  8251  omsuc  8253  onasuc  8255  oacl  8262  oneo  8309  omeulem1  8310  omeulem2  8311  oeordi  8315  oeworde  8321  oelim2  8323  oelimcl  8328  oeeulem  8329  oeeui  8330  oaabs2  8374  omxpenlem  8746  card2inf  9171  cantnflt  9287  cantnflem1d  9303  cnfcom  9315  r1ordg  9394  bndrank  9457  r1pw  9461  r1pwALT  9462  tcrank  9500  onssnum  9654  dfac12lem2  9758  cfsuc  9871  cfsmolem  9884  fin1a2lem1  10014  fin1a2lem2  10015  ttukeylem7  10129  alephreg  10196  gch2  10289  winainflem  10307  winalim2  10310  r1wunlim  10351  nqereu  10543  noextend  33606  noresle  33637  nosupno  33643  madeoldsuc  33804  ontgval  34357  ontgsucval  34358  onsuctop  34359  sucneqond  35273  onsetreclem2  46082