MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  topgele Structured version   Visualization version   GIF version

Theorem topgele 22852
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 22835 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 0opn 22826 . . . 4 (𝐽 ∈ Top β†’ βˆ… ∈ 𝐽)
31, 2syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆ… ∈ 𝐽)
4 toponmax 22848 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
53, 4prssd 4830 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ {βˆ…, 𝑋} βŠ† 𝐽)
6 toponuni 22836 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
7 eqimss2 4041 . . . 4 (𝑋 = βˆͺ 𝐽 β†’ βˆͺ 𝐽 βŠ† 𝑋)
86, 7syl 17 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 βŠ† 𝑋)
9 sspwuni 5107 . . 3 (𝐽 βŠ† 𝒫 𝑋 ↔ βˆͺ 𝐽 βŠ† 𝑋)
108, 9sylibr 233 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 βŠ† 𝒫 𝑋)
115, 10jca 510 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 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  βˆ…c0 4326  π’« cpw 4606  {cpr 4634  βˆͺ cuni 4912  β€˜cfv 6553  Topctop 22815  TopOnctopon 22832
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-top 22816  df-topon 22833
This theorem is referenced by:  topsn  22853  txindis  23558  dissneqlem  36852  ntrf2  43585
  Copyright terms: Public domain W3C validator