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Theorem ontgval 36414
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 22983 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 = (𝐵 ∩ 𝒫 𝑥))
2 inex1g 5325 . . . . . . 7 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3 onss 7804 . . . . . . . 8 (𝐵 ∈ On → 𝐵 ⊆ On)
4 ssinss1 4254 . . . . . . . 8 (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
53, 4syl 17 . . . . . . 7 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6 ssonuni 7799 . . . . . . 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 2827 . . . . . . 7 (𝑥 = (𝐵 ∩ 𝒫 𝑥) → (𝑥 ∈ On ↔ (𝐵 ∩ 𝒫 𝑥) ∈ On))
98biimprd 248 . . . . . 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 7808 . . . . . 6 (𝐵 ∈ On → 𝐵 ∈ On)
12 onsuc 7831 . . . . . 6 ( 𝐵 ∈ On → suc 𝐵 ∈ On)
1311, 12syl 17 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ On)
1410, 13jctird 526 . . . 4 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ∈ On ∧ suc 𝐵 ∈ On)))
15 tg1 22987 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
1615a1i 11 . . . . 5 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵))
17 sucidg 6467 . . . . . 6 ( 𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
1811, 17syl 17 . . . . 5 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
1916, 18jctird 526 . . . 4 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 𝐵 𝐵 ∈ suc 𝐵)))
20 ontr2 6433 . . . 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 6453 . . . 4 (𝑥 ∈ suc 𝐵 → (𝑥 𝐵𝑥 = 𝐵))
23 eloni 6396 . . . . . . . 8 (𝐵 ∈ On → Ord 𝐵)
24 orduniss 6483 . . . . . . . 8 (Ord 𝐵 𝐵𝐵)
2523, 24syl 17 . . . . . . 7 (𝐵 ∈ On → 𝐵𝐵)
26 bastg 22989 . . . . . . 7 (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
2725, 26sstrd 4006 . . . . . 6 (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
2827sseld 3994 . . . . 5 (𝐵 ∈ On → (𝑥 𝐵𝑥 ∈ (topGen‘𝐵)))
29 ssid 4018 . . . . . . 7 𝐵𝐵
30 eltg3i 22984 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵𝐵) → 𝐵 ∈ (topGen‘𝐵))
3129, 30mpan2 691 . . . . . 6 (𝐵 ∈ On → 𝐵 ∈ (topGen‘𝐵))
32 eleq1a 2834 . . . . . 6 ( 𝐵 ∈ (topGen‘𝐵) → (𝑥 = 𝐵𝑥 ∈ (topGen‘𝐵)))
3331, 32syl 17 . . . . 5 (𝐵 ∈ On → (𝑥 = 𝐵𝑥 ∈ (topGen‘𝐵)))
3428, 33jaod 859 . . . 4 (𝐵 ∈ On → ((𝑥 𝐵𝑥 = 𝐵) → 𝑥 ∈ (topGen‘𝐵)))
3522, 34syl5 34 . . 3 (𝐵 ∈ On → (𝑥 ∈ suc 𝐵𝑥 ∈ (topGen‘𝐵)))
3621, 35impbid 212 . 2 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ suc 𝐵))
3736eqrdv 2733 1 (𝐵 ∈ On → (topGen‘𝐵) = suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  Ord word 6385  Oncon0 6386  suc csuc 6388  cfv 6563  topGenctg 17484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-topgen 17490
This theorem is referenced by:  ontgsucval  36415
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