Theorem suceloniOLD 7807

Description: Obsolete version of onsuc 7806 as of 30-Nov-2024. (Contributed by NM, 6-Jun-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem suceloniOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelss 6405	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		velsn 4640	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		eqimss 4036	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
4	2, 3	sylbi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ⊆ 𝐴)
5	4	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥 ⊆ 𝐴))
6	1, 5	orim12d 963	. . . . . . 7 ⊢ (𝐴 ∈ On → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴)))
7		df-suc 6369	. . . . . . . . 9 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
8	7	eleq2i 2820	. . . . . . . 8 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴}))
9		elun 4144	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10	8, 9	bitr2i 276	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11		oridm 903	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴) ↔ 𝑥 ⊆ 𝐴)
12	6, 10, 11	3imtr3g 295	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴))
13		sssucid 6443	. . . . . 6 ⊢ 𝐴 ⊆ suc 𝐴
14		sstr2 3985	. . . . . 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 3140	. . . 4 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17		dftr3 5265	. . . 4 ⊢ (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
18	16, 17	sylibr 233	. . 3 ⊢ (𝐴 ∈ On → Tr suc 𝐴)
19		onss 7779	. . . . 5 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
20		snssi 4807	. . . . 5 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
21	19, 20	unssd 4182	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
22	7, 21	eqsstrid 4026	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ⊆ On)
23		ordon 7771	. . . 4 ⊢ Ord On
24		trssord 6380	. . . . 5 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25	24	3exp 1117	. . . 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 7800	. . 3 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ V)
29		elong 6371	. . 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 257	1 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 846   = wceq 1534   ∈ wcel 2099  ∀wral 3056  Vcvv 3469   ∪ cun 3942   ⊆ wss 3944  {csn 4624  Tr wtr 5259  Ord word 6362  Oncon0 6363  suc csuc 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-suc 6369

This theorem is referenced by: (None)