Theorem ontgval 31600

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 20575	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥))
2		inex1g 4729	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3		onss 6882	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		ssinss1 3803	. . . . . . . 8 ⊢ (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6		ssonuni 6878	. . . . . . 7 ⊢ ((𝐵 ∩ 𝒫 𝑥) ∈ V → ((𝐵 ∩ 𝒫 𝑥) ⊆ On → ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On))
7	2, 5, 6	sylc 63	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On)
8		eleq1 2676	. . . . . . 7 ⊢ (𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ∈ On ↔ ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On))
9	8	biimprd 237	. . . . . 6 ⊢ (𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥) → (∪ (𝐵 ∩ 𝒫 𝑥) ∈ On → 𝑥 ∈ On))
10	1, 7, 9	syl2imc 40	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ On))
11		onuni 6885	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ On)
12		suceloni 6905	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
13	11, 12	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
14	10, 13	jctird 565	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On)))
15		tg1 20579	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵))
17		sucidg 5720	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
18	11, 17	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
19	16, 18	jctird 565	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵)))
20		ontr2 5689	. . . 4 ⊢ ((𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On) → ((𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵) → 𝑥 ∈ suc ∪ 𝐵))
21	14, 19, 20	syl6c 68	. . 3 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ suc ∪ 𝐵))
22		elsuci 5708	. . . 4 ⊢ (𝑥 ∈ suc ∪ 𝐵 → (𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵))
23		eloni 5650	. . . . . . . 8 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		orduniss 5738	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ 𝐵 ⊆ 𝐵)
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ 𝐵)
26		bastg 20581	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
27	25, 26	sstrd 3578	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ (topGen‘𝐵))
28	27	sseld 3567	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
29		ssid 3587	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
30		eltg3i 20576	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐵 ⊆ 𝐵) → ∪ 𝐵 ∈ (topGen‘𝐵))
31	29, 30	mpan2 703	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ (topGen‘𝐵))
32		eleq1a 2683	. . . . . 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 394	. . . 4 ⊢ (𝐵 ∈ On → ((𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵) → 𝑥 ∈ (topGen‘𝐵)))
35	22, 34	syl5 33	. . 3 ⊢ (𝐵 ∈ On → (𝑥 ∈ suc ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
36	21, 35	impbid 201	. 2 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ suc ∪ 𝐵))
37	36	eqrdv 2608	1 ⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵)

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  Ord word 5639  Oncon0 5640  suc csuc 5642  ‘cfv 5804  topGenctg 15921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927

This theorem is referenced by:  ontgsucval  31601