Theorem ontgval 36666

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 22950	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥))
2		inex1g 5254	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3		onss 7735	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		ssinss1 4181	. . . . . . . 8 ⊢ (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6		ssonuni 7730	. . . . . . 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 2828	. . . . . . 7 ⊢ (𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ∈ On ↔ ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On))
9	8	biimprd 249	. . . . . 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 7738	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ On)
12		onsuc 7760	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
13	11, 12	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
14	10, 13	jctird 531	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On)))
15		tg1 22954	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵))
17		sucidg 6400	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
18	11, 17	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
19	16, 18	jctird 531	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵)))
20		ontr2 6365	. . . 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 6386	. . . 4 ⊢ (𝑥 ∈ suc ∪ 𝐵 → (𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵))
23		eloni 6327	. . . . . . . 8 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		orduniss 6416	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ 𝐵 ⊆ 𝐵)
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ 𝐵)
26		bastg 22956	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
27	25, 26	sstrd 3932	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ (topGen‘𝐵))
28	27	sseld 3921	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
29		ssid 3944	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
30		eltg3i 22951	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐵 ⊆ 𝐵) → ∪ 𝐵 ∈ (topGen‘𝐵))
31	29, 30	mpan2 697	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ (topGen‘𝐵))
32		eleq1a 2835	. . . . . 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 865	. . . 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 213	. 2 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ suc ∪ 𝐵))
37	36	eqrdv 2738	1 ⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4536  ∪ cuni 4845  Ord word 6316  Oncon0 6317  suc csuc 6319  ‘cfv 6492  topGenctg 17398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fv 6500  df-topgen 17404

This theorem is referenced by:  ontgsucval  36667