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Theorem topgele 21063
Description: The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
topgele (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))

Proof of Theorem topgele
StepHypRef Expression
1 topontop 21046 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 0opn 21037 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
4 toponmax 21059 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 0ex 4984 . . . 4 ∅ ∈ V
6 prssg 4538 . . . 4 ((∅ ∈ V ∧ 𝑋𝐽) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
75, 4, 6sylancr 582 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
83, 4, 7mpbi2and 704 . 2 (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽)
9 toponuni 21047 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
10 eqimss2 3854 . . . 4 (𝑋 = 𝐽 𝐽𝑋)
119, 10syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽𝑋)
12 sspwuni 4802 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
1311, 12sylibr 226 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
148, 13jca 508 1 (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  wss 3769  c0 4115  𝒫 cpw 4349  {cpr 4370   cuni 4628  cfv 6101  Topctop 21026  TopOnctopon 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-top 21027  df-topon 21044
This theorem is referenced by:  topsn  21064  txindis  21766  dissneqlem  33686  ntrf2  39204
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