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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 21515 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | 0opn 21506 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽) |
4 | toponmax 21528 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
5 | 3, 4 | prssd 4748 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽) |
6 | toponuni 21516 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
7 | eqimss2 4023 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝑋) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ⊆ 𝑋) |
9 | sspwuni 5014 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽 ⊆ 𝑋) | |
10 | 8, 9 | sylibr 236 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋) |
11 | 5, 10 | jca 514 | 1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 {cpr 4562 ∪ cuni 4831 ‘cfv 6349 Topctop 21495 TopOnctopon 21512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-top 21496 df-topon 21513 |
This theorem is referenced by: topsn 21533 txindis 22236 dissneqlem 34615 ntrf2 40467 |
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