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Theorem topgele 22295
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 22278 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 0opn 22269 . . . 4 (𝐽 ∈ Top β†’ βˆ… ∈ 𝐽)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆ… ∈ 𝐽)
4 toponmax 22291 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
53, 4prssd 4783 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ {βˆ…, 𝑋} βŠ† 𝐽)
6 toponuni 22279 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
7 eqimss2 4002 . . . 4 (𝑋 = βˆͺ 𝐽 β†’ βˆͺ 𝐽 βŠ† 𝑋)
86, 7syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 βŠ† 𝑋)
9 sspwuni 5061 . . 3 (𝐽 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝐽 βŠ† 𝑋)
108, 9sylibr 233 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 βŠ† 𝒫 𝑋)
115, 10jca 513 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 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  βˆ…c0 4283  π’« cpw 4561  {cpr 4589  βˆͺ cuni 4866  β€˜cfv 6497  Topctop 22258  TopOnctopon 22275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-top 22259  df-topon 22276
This theorem is referenced by:  topsn  22296  txindis  23001  dissneqlem  35857  ntrf2  42484
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