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Theorem topgele 22783
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 ∈ (TopOnβ€˜π‘‹) β†’ ({βˆ…, 𝑋} βŠ† 𝐽 ∧ 𝐽 βŠ† 𝒫 𝑋))

Proof of Theorem topgele
StepHypRef Expression
1 topontop 22766 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 0opn 22757 . . . 4 (𝐽 ∈ Top β†’ βˆ… ∈ 𝐽)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆ… ∈ 𝐽)
4 toponmax 22779 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
53, 4prssd 4820 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ {βˆ…, 𝑋} βŠ† 𝐽)
6 toponuni 22767 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
7 eqimss2 4036 . . . 4 (𝑋 = βˆͺ 𝐽 β†’ βˆͺ 𝐽 βŠ† 𝑋)
86, 7syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 βŠ† 𝑋)
9 sspwuni 5096 . . 3 (𝐽 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝐽 βŠ† 𝑋)
108, 9sylibr 233 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 βŠ† 𝒫 𝑋)
115, 10jca 511 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ ({βˆ…, 𝑋} βŠ† 𝐽 ∧ 𝐽 βŠ† 𝒫 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  {cpr 4625  βˆͺ cuni 4902  β€˜cfv 6536  Topctop 22746  TopOnctopon 22763
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-top 22747  df-topon 22764
This theorem is referenced by:  topsn  22784  txindis  23489  dissneqlem  36728  ntrf2  43432
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