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Theorem topgele 20549
 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 20541 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 0opn 20534 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
4 toponmax 20543 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 0ex 4718 . . . 4 ∅ ∈ V
6 prssg 4290 . . . 4 ((∅ ∈ V ∧ 𝑋𝐽) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
75, 4, 6sylancr 694 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
83, 4, 7mpbi2and 958 . 2 (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽)
9 toponuni 20542 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
10 eqimss2 3621 . . . 4 (𝑋 = 𝐽 𝐽𝑋)
119, 10syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽𝑋)
12 sspwuni 4547 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
1311, 12sylibr 223 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
148, 13jca 553 1 (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {cpr 4127  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  TopOnctopon 20518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523 This theorem is referenced by:  topsn  20550  txindis  21247  dissneqlem  32363  ntrf2  37442
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