Theorem ontgval 36419

Description: The topology generated from an ordinal number 𝐵 is suc ∪ 𝐵. (Contributed by Chen-Pang He, 10-Oct-2015.)

Assertion

Ref	Expression
ontgval	⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵)

Proof of Theorem ontgval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eltg4i 22847	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥))
2		inex1g 5274	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3		onss 7761	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		ssinss1 4209	. . . . . . . 8 ⊢ (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6		ssonuni 7756	. . . . . . 7 ⊢ ((𝐵 ∩ 𝒫 𝑥) ∈ V → ((𝐵 ∩ 𝒫 𝑥) ⊆ On → ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On))
7	2, 5, 6	sylc 65	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On)
8		eleq1 2816	. . . . . . 7 ⊢ (𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ∈ On ↔ ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On))
9	8	biimprd 248	. . . . . 6 ⊢ (𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥) → (∪ (𝐵 ∩ 𝒫 𝑥) ∈ On → 𝑥 ∈ On))
10	1, 7, 9	syl2imc 41	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ On))
11		onuni 7764	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ On)
12		onsuc 7787	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
13	11, 12	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
14	10, 13	jctird 526	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On)))
15		tg1 22851	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵))
17		sucidg 6415	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
18	11, 17	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
19	16, 18	jctird 526	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵)))
20		ontr2 6380	. . . 4 ⊢ ((𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On) → ((𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵) → 𝑥 ∈ suc ∪ 𝐵))
21	14, 19, 20	syl6c 70	. . 3 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ suc ∪ 𝐵))
22		elsuci 6401	. . . 4 ⊢ (𝑥 ∈ suc ∪ 𝐵 → (𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵))
23		eloni 6342	. . . . . . . 8 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		orduniss 6431	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ 𝐵 ⊆ 𝐵)
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ 𝐵)
26		bastg 22853	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
27	25, 26	sstrd 3957	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ (topGen‘𝐵))
28	27	sseld 3945	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
29		ssid 3969	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
30		eltg3i 22848	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐵 ⊆ 𝐵) → ∪ 𝐵 ∈ (topGen‘𝐵))
31	29, 30	mpan2 691	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ (topGen‘𝐵))
32		eleq1a 2823	. . . . . 6 ⊢ (∪ 𝐵 ∈ (topGen‘𝐵) → (𝑥 = ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
33	31, 32	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 = ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
34	28, 33	jaod 859	. . . 4 ⊢ (𝐵 ∈ On → ((𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵) → 𝑥 ∈ (topGen‘𝐵)))
35	22, 34	syl5 34	. . 3 ⊢ (𝐵 ∈ On → (𝑥 ∈ suc ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
36	21, 35	impbid 212	. 2 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ suc ∪ 𝐵))
37	36	eqrdv 2727	1 ⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  𝒫 cpw 4563  ∪ cuni 4871  Ord word 6331  Oncon0 6332  suc csuc 6334  ‘cfv 6511  topGenctg 17400

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-ord 6335  df-on 6336  df-suc 6338  df-iota 6464  df-fun 6513  df-fv 6519  df-topgen 17406

This theorem is referenced by:  ontgsucval  36420