Theorem suceloni 7508

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 6201	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		velsn 4541	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		eqimss 3971	. . . . . . . . . 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 962	. . . . . . 7 ⊢ (𝐴 ∈ On → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴)))
7		df-suc 6165	. . . . . . . . 9 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
8	7	eleq2i 2881	. . . . . . . 8 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴}))
9		elun 4076	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10	8, 9	bitr2i 279	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11		oridm 902	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴) ↔ 𝑥 ⊆ 𝐴)
12	6, 10, 11	3imtr3g 298	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴))
13		sssucid 6236	. . . . . 6 ⊢ 𝐴 ⊆ suc 𝐴
14		sstr2 3922	. . . . . 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 3148	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17		dftr3 5140	. . . 4 ⊢ (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
18	16, 17	sylibr 237	. . 3 ⊢ (𝐴 ∈ On → Tr suc 𝐴)
19		onss 7485	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
20		snssi 4701	. . . . 5 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
21	19, 20	unssd 4113	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
22	7, 21	eqsstrid 3963	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ⊆ On)
23		ordon 7478	. . . 4 ⊢ Ord On
24		trssord 6176	. . . . 5 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25	24	3exp 1116	. . . 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 7505	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
29		elong 6167	. . 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 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  {csn 4525  Tr wtr 5136  Ord word 6158  Oncon0 6159  suc csuc 6161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-suc 6165

This theorem is referenced by:  ordsuc  7509  unon  7526  onsuci  7533  ordunisuc2  7539  ordzsl  7540  onzsl  7541  tfindsg  7555  dfom2  7562  findsg  7590  tfrlem12  8008  oasuc  8132  omsuc  8134  onasuc  8136  oacl  8143  oneo  8190  omeulem1  8191  omeulem2  8192  oeordi  8196  oeworde  8202  oelim2  8204  oelimcl  8209  oeeulem  8210  oeeui  8211  oaabs2  8255  omxpenlem  8601  card2inf  9003  cantnflt  9119  cantnflem1d  9135  cnfcom  9147  r1ordg  9191  bndrank  9254  r1pw  9258  r1pwALT  9259  tcrank  9297  onssnum  9451  dfac12lem2  9555  cfsuc  9668  cfsmolem  9681  fin1a2lem1  9811  fin1a2lem2  9812  ttukeylem7  9926  alephreg  9993  gch2  10086  winainflem  10104  winalim2  10107  r1wunlim  10148  nqereu  10340  noextend  33286  noresle  33313  nosupno  33316  ontgval  33892  ontgsucval  33893  onsuctop  33894  sucneqond  34782  onsetreclem2  45235