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Mirrors > Home > MPE Home > Th. List > topgele | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
topgele | ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 21518 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | 0opn 21509 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽) |
4 | toponmax 21531 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
5 | 3, 4 | prssd 4715 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽) |
6 | toponuni 21519 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
7 | eqimss2 3972 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝑋) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ⊆ 𝑋) |
9 | sspwuni 4985 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽 ⊆ 𝑋) | |
10 | 8, 9 | sylibr 237 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋) |
11 | 5, 10 | jca 515 | 1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {cpr 4527 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 TopOnctopon 21515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-top 21499 df-topon 21516 |
This theorem is referenced by: topsn 21536 txindis 22239 dissneqlem 34757 ntrf2 40827 |
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