Theorem ontgval 33779

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 21568	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥))
2		inex1g 5223	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3		onss 7505	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		ssinss1 4214	. . . . . . . 8 ⊢ (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6		ssonuni 7501	. . . . . . 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 2900	. . . . . . 7 ⊢ (𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ∈ On ↔ ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On))
9	8	biimprd 250	. . . . . 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 7508	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ On)
12		suceloni 7528	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
13	11, 12	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
14	10, 13	jctird 529	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On)))
15		tg1 21572	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵))
17		sucidg 6269	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
18	11, 17	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
19	16, 18	jctird 529	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵)))
20		ontr2 6238	. . . 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 6257	. . . 4 ⊢ (𝑥 ∈ suc ∪ 𝐵 → (𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵))
23		eloni 6201	. . . . . . . 8 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		orduniss 6285	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ 𝐵 ⊆ 𝐵)
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ 𝐵)
26		bastg 21574	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
27	25, 26	sstrd 3977	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ (topGen‘𝐵))
28	27	sseld 3966	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
29		ssid 3989	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
30		eltg3i 21569	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐵 ⊆ 𝐵) → ∪ 𝐵 ∈ (topGen‘𝐵))
31	29, 30	mpan2 689	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ (topGen‘𝐵))
32		eleq1a 2908	. . . . . 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 855	. . . 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 214	. 2 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ suc ∪ 𝐵))
37	36	eqrdv 2819	1 ⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  ∪ cuni 4838  Ord word 6190  Oncon0 6191  suc csuc 6193  ‘cfv 6355  topGenctg 16711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6314  df-fun 6357  df-fv 6363  df-topgen 16717

This theorem is referenced by:  ontgsucval  33780