Theorem ontgval 33892

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 21565	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥))
2		inex1g 5187	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3		onss 7485	. . . . . . . 8 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		ssinss1 4164	. . . . . . . 8 ⊢ (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6		ssonuni 7481	. . . . . . 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 2877	. . . . . . 7 ⊢ (𝑥 = ∪ (𝐵 ∩ 𝒫 𝑥) → (𝑥 ∈ On ↔ ∪ (𝐵 ∩ 𝒫 𝑥) ∈ On))
9	8	biimprd 251	. . . . . 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 7488	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ On)
12		suceloni 7508	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
13	11, 12	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → suc ∪ 𝐵 ∈ On)
14	10, 13	jctird 530	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ∈ On ∧ suc ∪ 𝐵 ∈ On)))
15		tg1 21569	. . . . . 6 ⊢ (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵)
16	15	a1i 11	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ⊆ ∪ 𝐵))
17		sucidg 6237	. . . . . 6 ⊢ (∪ 𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
18	11, 17	syl 17	. . . . 5 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ suc ∪ 𝐵)
19	16, 18	jctird 530	. . . 4 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ∈ suc ∪ 𝐵)))
20		ontr2 6206	. . . 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 6225	. . . 4 ⊢ (𝑥 ∈ suc ∪ 𝐵 → (𝑥 ∈ ∪ 𝐵 ∨ 𝑥 = ∪ 𝐵))
23		eloni 6169	. . . . . . . 8 ⊢ (𝐵 ∈ On → Ord 𝐵)
24		orduniss 6253	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ 𝐵 ⊆ 𝐵)
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ 𝐵)
26		bastg 21571	. . . . . . 7 ⊢ (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
27	25, 26	sstrd 3925	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ⊆ (topGen‘𝐵))
28	27	sseld 3914	. . . . 5 ⊢ (𝐵 ∈ On → (𝑥 ∈ ∪ 𝐵 → 𝑥 ∈ (topGen‘𝐵)))
29		ssid 3937	. . . . . . 7 ⊢ 𝐵 ⊆ 𝐵
30		eltg3i 21566	. . . . . . 7 ⊢ ((𝐵 ∈ On ∧ 𝐵 ⊆ 𝐵) → ∪ 𝐵 ∈ (topGen‘𝐵))
31	29, 30	mpan2 690	. . . . . 6 ⊢ (𝐵 ∈ On → ∪ 𝐵 ∈ (topGen‘𝐵))
32		eleq1a 2885	. . . . . 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 856	. . . 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 215	. 2 ⊢ (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ suc ∪ 𝐵))
37	36	eqrdv 2796	1 ⊢ (𝐵 ∈ On → (topGen‘𝐵) = suc ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  Ord word 6158  Oncon0 6159  suc csuc 6161  ‘cfv 6324  topGenctg 16703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fv 6332  df-topgen 16709

This theorem is referenced by:  ontgsucval  33893