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Theorem topgele 20715
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 20699 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 0opn 20690 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
4 toponmax 20711 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 0ex 4781 . . . 4 ∅ ∈ V
6 prssg 4341 . . . 4 ((∅ ∈ V ∧ 𝑋𝐽) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
75, 4, 6sylancr 694 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
83, 4, 7mpbi2and 955 . 2 (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽)
9 toponuni 20700 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
10 eqimss2 3650 . . . 4 (𝑋 = 𝐽 𝐽𝑋)
119, 10syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽𝑋)
12 sspwuni 4602 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
1311, 12sylibr 224 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
148, 13jca 554 1 (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  wss 3567  c0 3907  𝒫 cpw 4149  {cpr 4170   cuni 4427  cfv 5876  Topctop 20679  TopOnctopon 20696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-top 20680  df-topon 20697
This theorem is referenced by:  topsn  20716  txindis  21418  dissneqlem  33158  ntrf2  38242
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