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Theorem ontgval 32125
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.
StepHypRef Expression
1 eltg4i 20704 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 = (𝐵 ∩ 𝒫 𝑥))
2 inex1g 4771 . . . . . . 7 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3 onss 6952 . . . . . . . 8 (𝐵 ∈ On → 𝐵 ⊆ On)
4 ssinss1 3825 . . . . . . . 8 (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
53, 4syl 17 . . . . . . 7 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6 ssonuni 6948 . . . . . . 7 ((𝐵 ∩ 𝒫 𝑥) ∈ V → ((𝐵 ∩ 𝒫 𝑥) ⊆ On → (𝐵 ∩ 𝒫 𝑥) ∈ On))
72, 5, 6sylc 65 . . . . . 6 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ On)
8 eleq1 2686 . . . . . . 7 (𝑥 = (𝐵 ∩ 𝒫 𝑥) → (𝑥 ∈ On ↔ (𝐵 ∩ 𝒫 𝑥) ∈ On))
98biimprd 238 . . . . . 6 (𝑥 = (𝐵 ∩ 𝒫 𝑥) → ( (𝐵 ∩ 𝒫 𝑥) ∈ On → 𝑥 ∈ On))
101, 7, 9syl2imc 41 . . . . 5 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ On))
11 onuni 6955 . . . . . 6 (𝐵 ∈ On → 𝐵 ∈ On)
12 suceloni 6975 . . . . . 6 ( 𝐵 ∈ On → suc 𝐵 ∈ On)
1311, 12syl 17 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ On)
1410, 13jctird 566 . . . 4 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ∈ On ∧ suc 𝐵 ∈ On)))
15 tg1 20708 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
1615a1i 11 . . . . 5 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵))
17 sucidg 5772 . . . . . 6 ( 𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
1811, 17syl 17 . . . . 5 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
1916, 18jctird 566 . . . 4 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 𝐵 𝐵 ∈ suc 𝐵)))
20 ontr2 5741 . . . 4 ((𝑥 ∈ On ∧ suc 𝐵 ∈ On) → ((𝑥 𝐵 𝐵 ∈ suc 𝐵) → 𝑥 ∈ suc 𝐵))
2114, 19, 20syl6c 70 . . 3 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 ∈ suc 𝐵))
22 elsuci 5760 . . . 4 (𝑥 ∈ suc 𝐵 → (𝑥 𝐵𝑥 = 𝐵))
23 eloni 5702 . . . . . . . 8 (𝐵 ∈ On → Ord 𝐵)
24 orduniss 5790 . . . . . . . 8 (Ord 𝐵 𝐵𝐵)
2523, 24syl 17 . . . . . . 7 (𝐵 ∈ On → 𝐵𝐵)
26 bastg 20710 . . . . . . 7 (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
2725, 26sstrd 3598 . . . . . 6 (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
2827sseld 3587 . . . . 5 (𝐵 ∈ On → (𝑥 𝐵𝑥 ∈ (topGen‘𝐵)))
29 ssid 3609 . . . . . . 7 𝐵𝐵
30 eltg3i 20705 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵𝐵) → 𝐵 ∈ (topGen‘𝐵))
3129, 30mpan2 706 . . . . . 6 (𝐵 ∈ On → 𝐵 ∈ (topGen‘𝐵))
32 eleq1a 2693 . . . . . 6 ( 𝐵 ∈ (topGen‘𝐵) → (𝑥 = 𝐵𝑥 ∈ (topGen‘𝐵)))
3331, 32syl 17 . . . . 5 (𝐵 ∈ On → (𝑥 = 𝐵𝑥 ∈ (topGen‘𝐵)))
3428, 33jaod 395 . . . 4 (𝐵 ∈ On → ((𝑥 𝐵𝑥 = 𝐵) → 𝑥 ∈ (topGen‘𝐵)))
3522, 34syl5 34 . . 3 (𝐵 ∈ On → (𝑥 ∈ suc 𝐵𝑥 ∈ (topGen‘𝐵)))
3621, 35impbid 202 . 2 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ suc 𝐵))
3736eqrdv 2619 1 (𝐵 ∈ On → (topGen‘𝐵) = suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  cin 3559  wss 3560  𝒫 cpw 4136   cuni 4409  Ord word 5691  Oncon0 5692  suc csuc 5694  cfv 5857  topGenctg 16038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-ord 5695  df-on 5696  df-suc 5698  df-iota 5820  df-fun 5859  df-fv 5865  df-topgen 16044
This theorem is referenced by:  ontgsucval  32126
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