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Theorem ontgval 32555
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 20812 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 = (𝐵 ∩ 𝒫 𝑥))
2 inex1g 4834 . . . . . . 7 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3 onss 7032 . . . . . . . 8 (𝐵 ∈ On → 𝐵 ⊆ On)
4 ssinss1 3874 . . . . . . . 8 (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
53, 4syl 17 . . . . . . 7 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6 ssonuni 7028 . . . . . . 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 2718 . . . . . . 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 7035 . . . . . 6 (𝐵 ∈ On → 𝐵 ∈ On)
12 suceloni 7055 . . . . . 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 20816 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
1615a1i 11 . . . . 5 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵))
17 sucidg 5841 . . . . . 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 5810 . . . 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 5829 . . . 4 (𝑥 ∈ suc 𝐵 → (𝑥 𝐵𝑥 = 𝐵))
23 eloni 5771 . . . . . . . 8 (𝐵 ∈ On → Ord 𝐵)
24 orduniss 5859 . . . . . . . 8 (Ord 𝐵 𝐵𝐵)
2523, 24syl 17 . . . . . . 7 (𝐵 ∈ On → 𝐵𝐵)
26 bastg 20818 . . . . . . 7 (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
2725, 26sstrd 3646 . . . . . 6 (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
2827sseld 3635 . . . . 5 (𝐵 ∈ On → (𝑥 𝐵𝑥 ∈ (topGen‘𝐵)))
29 ssid 3657 . . . . . . 7 𝐵𝐵
30 eltg3i 20813 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵𝐵) → 𝐵 ∈ (topGen‘𝐵))
3129, 30mpan2 707 . . . . . 6 (𝐵 ∈ On → 𝐵 ∈ (topGen‘𝐵))
32 eleq1a 2725 . . . . . 6 ( 𝐵 ∈ (topGen‘𝐵) → (𝑥 = 𝐵𝑥 ∈ (topGen‘𝐵)))
3331, 32syl 17 . . . . 5 (𝐵 ∈ On → (𝑥 = 𝐵𝑥 ∈ (topGen‘𝐵)))
3428, 33jaod 394 . . . 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 2649 1 (𝐵 ∈ On → (topGen‘𝐵) = suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468  Ord word 5760  Oncon0 5761  suc csuc 5763  cfv 5926  topGenctg 16145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151
This theorem is referenced by:  ontgsucval  32556
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