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Theorem topgele 22881
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 22864 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 0opn 22855 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
4 toponmax 22877 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
53, 4prssd 4827 . 2 (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽)
6 toponuni 22865 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
7 eqimss2 4036 . . . 4 (𝑋 = 𝐽 𝐽𝑋)
86, 7syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽𝑋)
9 sspwuni 5104 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
108, 9sylibr 233 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
115, 10jca 510 1 (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wss 3944  c0 4322  𝒫 cpw 4604  {cpr 4632   cuni 4909  cfv 6549  Topctop 22844  TopOnctopon 22861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-top 22845  df-topon 22862
This theorem is referenced by:  topsn  22882  txindis  23587  dissneqlem  36952  ntrf2  43698
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