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Theorem topgele 12233
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 12218 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 0opn 12210 . . . 4 (𝐽 ∈ Top → ∅ ∈ 𝐽)
31, 2syl 14 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽)
4 toponmax 12229 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 0ex 4062 . . . 4 ∅ ∈ V
6 prssg 3684 . . . 4 ((∅ ∈ V ∧ 𝑋𝐽) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
75, 4, 6sylancr 411 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((∅ ∈ 𝐽𝑋𝐽) ↔ {∅, 𝑋} ⊆ 𝐽))
83, 4, 7mpbi2and 928 . 2 (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽)
9 toponuni 12219 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
10 eqimss2 3156 . . . 4 (𝑋 = 𝐽 𝐽𝑋)
119, 10syl 14 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽𝑋)
12 sspwuni 3904 . . 3 (𝐽 ⊆ 𝒫 𝑋 𝐽𝑋)
1311, 12sylibr 133 . 2 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
148, 13jca 304 1 (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽𝐽 ⊆ 𝒫 𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  Vcvv 2689  wss 3075  c0 3367  𝒫 cpw 3514  {cpr 3532   cuni 3743  cfv 5130  Topctop 12201  TopOnctopon 12214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-top 12202  df-topon 12215
This theorem is referenced by: (None)
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