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Theorem topgele 20857
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 20841 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 0opn 20832 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
4 toponmax 20853 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 0ex 4898 . . . 4 ∅ ∈ V
6 prssg 4458 . . . 4 ((∅ ∈ V ∧ 𝑋𝐽) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
75, 4, 6sylancr 698 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
83, 4, 7mpbi2and 994 . 2 (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽)
9 toponuni 20842 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
10 eqimss2 3764 . . . 4 (𝑋 = 𝐽 𝐽𝑋)
119, 10syl 17 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽𝑋)
12 sspwuni 4719 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
1311, 12sylibr 224 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
148, 13jca 555 1 (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  Vcvv 3304  wss 3680  c0 4023  𝒫 cpw 4266  {cpr 4287   cuni 4544  cfv 6001  Topctop 20821  TopOnctopon 20838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-top 20822  df-topon 20839
This theorem is referenced by:  topsn  20858  txindis  21560  dissneqlem  33419  ntrf2  38841
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