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| Mirrors > Home > ILE Home > Th. List > topgele | 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 14647 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
| 2 | 0opn 14639 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
| 3 | 1, 2 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽) |
| 4 | toponmax 14658 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
| 5 | 0ex 4188 | . . . 4 ⊢ ∅ ∈ V | |
| 6 | prssg 3802 | . . . 4 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝐽) → ((∅ ∈ 𝐽 ∧ 𝑋 ∈ 𝐽) ↔ {∅, 𝑋} ⊆ 𝐽)) | |
| 7 | 5, 4, 6 | sylancr 414 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((∅ ∈ 𝐽 ∧ 𝑋 ∈ 𝐽) ↔ {∅, 𝑋} ⊆ 𝐽)) |
| 8 | 3, 4, 7 | mpbi2and 946 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽) |
| 9 | toponuni 14648 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
| 10 | eqimss2 3257 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝑋) | |
| 11 | 9, 10 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ⊆ 𝑋) |
| 12 | sspwuni 4027 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽 ⊆ 𝑋) | |
| 13 | 11, 12 | sylibr 134 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ⊆ 𝒫 𝑋) |
| 14 | 8, 13 | jca 306 | 1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ({∅, 𝑋} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 𝑋)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1373 ∈ wcel 2178 Vcvv 2777 ⊆ wss 3175 ∅c0 3469 𝒫 cpw 3627 {cpr 3645 ∪ cuni 3865 ‘cfv 5291 Topctop 14630 TopOnctopon 14643 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-sep 4179 ax-nul 4187 ax-pow 4235 ax-pr 4270 ax-un 4499 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ral 2491 df-rex 2492 df-rab 2495 df-v 2779 df-sbc 3007 df-dif 3177 df-un 3179 df-in 3181 df-ss 3188 df-nul 3470 df-pw 3629 df-sn 3650 df-pr 3651 df-op 3653 df-uni 3866 df-br 4061 df-opab 4123 df-mpt 4124 df-id 4359 df-xp 4700 df-rel 4701 df-cnv 4702 df-co 4703 df-dm 4704 df-iota 5252 df-fun 5293 df-fv 5299 df-top 14631 df-topon 14644 |
| This theorem is referenced by: (None) |
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