Theorem sucexeloniOLD 7766

Description: Obsolete version of sucexeloni 7765 as of 6-Jan-2025. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sucexeloniOLD	⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Proof of Theorem sucexeloniOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onelss 6362	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		velsn 4601	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3		eqimss 4002	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴)
4	2, 3	sylbi 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ⊆ 𝐴)
5	4	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝑥 ∈ {𝐴} → 𝑥 ⊆ 𝐴))
6	1, 5	orim12d 966	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) → (𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴)))
7		df-suc 6326	. . . . . . . . . 10 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
8	7	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑥 ∈ suc 𝐴 ↔ 𝑥 ∈ (𝐴 ∪ {𝐴}))
9		elun 4112	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10	8, 9	bitr2i 276	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) ↔ 𝑥 ∈ suc 𝐴)
11		oridm 904	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∨ 𝑥 ⊆ 𝐴) ↔ 𝑥 ⊆ 𝐴)
12	6, 10, 11	3imtr3g 295	. . . . . . 7 ⊢ (𝐴 ∈ On → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴))
13		sssucid 6402	. . . . . . 7 ⊢ 𝐴 ⊆ suc 𝐴
14		sstr2 3950	. . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ suc 𝐴 → 𝑥 ⊆ suc 𝐴))
15	12, 13, 14	syl6mpi 67	. . . . . 6 ⊢ (𝐴 ∈ On → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ suc 𝐴))
16	15	ralrimiv 3124	. . . . 5 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
17		dftr3 5215	. . . . 5 ⊢ (Tr suc 𝐴 ↔ ∀𝑥 ∈ suc 𝐴𝑥 ⊆ suc 𝐴)
18	16, 17	sylibr 234	. . . 4 ⊢ (𝐴 ∈ On → Tr suc 𝐴)
19		onss 7741	. . . . . 6 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
20		snssi 4768	. . . . . 6 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
21	19, 20	unssd 4151	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ∪ {𝐴}) ⊆ On)
22	7, 21	eqsstrid 3982	. . . 4 ⊢ (𝐴 ∈ On → suc 𝐴 ⊆ On)
23		ordon 7733	. . . . 5 ⊢ Ord On
24		trssord 6337	. . . . . 6 ⊢ ((Tr suc 𝐴 ∧ suc 𝐴 ⊆ On ∧ Ord On) → Ord suc 𝐴)
25	24	3exp 1119	. . . . 5 ⊢ (Tr suc 𝐴 → (suc 𝐴 ⊆ On → (Ord On → Ord suc 𝐴)))
26	23, 25	mpii 46	. . . 4 ⊢ (Tr suc 𝐴 → (suc 𝐴 ⊆ On → Ord suc 𝐴))
27	18, 22, 26	sylc 65	. . 3 ⊢ (𝐴 ∈ On → Ord suc 𝐴)
28	27	adantr 480	. 2 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → Ord suc 𝐴)
29		elong 6328	. . 3 ⊢ (suc 𝐴 ∈ 𝑉 → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
30	29	adantl 481	. 2 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
31	28, 30	mpbird 257	1 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∪ cun 3909   ⊆ wss 3911  {csn 4585  Tr wtr 5209  Ord word 6319  Oncon0 6320  suc csuc 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324  df-suc 6326

This theorem is referenced by: (None)