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Theorem ontgval 33783
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 21571 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 = (𝐵 ∩ 𝒫 𝑥))
2 inex1g 5226 . . . . . . 7 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ∈ V)
3 onss 7508 . . . . . . . 8 (𝐵 ∈ On → 𝐵 ⊆ On)
4 ssinss1 4217 . . . . . . . 8 (𝐵 ⊆ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
53, 4syl 17 . . . . . . 7 (𝐵 ∈ On → (𝐵 ∩ 𝒫 𝑥) ⊆ On)
6 ssonuni 7504 . . . . . . 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 2903 . . . . . . 7 (𝑥 = (𝐵 ∩ 𝒫 𝑥) → (𝑥 ∈ On ↔ (𝐵 ∩ 𝒫 𝑥) ∈ On))
98biimprd 250 . . . . . 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 7511 . . . . . 6 (𝐵 ∈ On → 𝐵 ∈ On)
12 suceloni 7531 . . . . . 6 ( 𝐵 ∈ On → suc 𝐵 ∈ On)
1311, 12syl 17 . . . . 5 (𝐵 ∈ On → suc 𝐵 ∈ On)
1410, 13jctird 529 . . . 4 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 ∈ On ∧ suc 𝐵 ∈ On)))
15 tg1 21575 . . . . . 6 (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵)
1615a1i 11 . . . . 5 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → 𝑥 𝐵))
17 sucidg 6272 . . . . . 6 ( 𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
1811, 17syl 17 . . . . 5 (𝐵 ∈ On → 𝐵 ∈ suc 𝐵)
1916, 18jctird 529 . . . 4 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) → (𝑥 𝐵 𝐵 ∈ suc 𝐵)))
20 ontr2 6241 . . . 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 6260 . . . 4 (𝑥 ∈ suc 𝐵 → (𝑥 𝐵𝑥 = 𝐵))
23 eloni 6204 . . . . . . . 8 (𝐵 ∈ On → Ord 𝐵)
24 orduniss 6288 . . . . . . . 8 (Ord 𝐵 𝐵𝐵)
2523, 24syl 17 . . . . . . 7 (𝐵 ∈ On → 𝐵𝐵)
26 bastg 21577 . . . . . . 7 (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
2725, 26sstrd 3980 . . . . . 6 (𝐵 ∈ On → 𝐵 ⊆ (topGen‘𝐵))
2827sseld 3969 . . . . 5 (𝐵 ∈ On → (𝑥 𝐵𝑥 ∈ (topGen‘𝐵)))
29 ssid 3992 . . . . . . 7 𝐵𝐵
30 eltg3i 21572 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐵𝐵) → 𝐵 ∈ (topGen‘𝐵))
3129, 30mpan2 689 . . . . . 6 (𝐵 ∈ On → 𝐵 ∈ (topGen‘𝐵))
32 eleq1a 2911 . . . . . 6 ( 𝐵 ∈ (topGen‘𝐵) → (𝑥 = 𝐵𝑥 ∈ (topGen‘𝐵)))
3331, 32syl 17 . . . . 5 (𝐵 ∈ On → (𝑥 = 𝐵𝑥 ∈ (topGen‘𝐵)))
3428, 33jaod 855 . . . 4 (𝐵 ∈ On → ((𝑥 𝐵𝑥 = 𝐵) → 𝑥 ∈ (topGen‘𝐵)))
3522, 34syl5 34 . . 3 (𝐵 ∈ On → (𝑥 ∈ suc 𝐵𝑥 ∈ (topGen‘𝐵)))
3621, 35impbid 214 . 2 (𝐵 ∈ On → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ∈ suc 𝐵))
3736eqrdv 2822 1 (𝐵 ∈ On → (topGen‘𝐵) = suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wo 843   = wceq 1536  wcel 2113  Vcvv 3497  cin 3938  wss 3939  𝒫 cpw 4542   cuni 4841  Ord word 6193  Oncon0 6194  suc csuc 6196  cfv 6358  topGenctg 16714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-ord 6197  df-on 6198  df-suc 6200  df-iota 6317  df-fun 6360  df-fv 6366  df-topgen 16720
This theorem is referenced by:  ontgsucval  33784
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