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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 14728 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
| 2 | 0opn 14720 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
| 3 | 1, 2 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽) |
| 4 | toponmax 14739 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
| 5 | 0ex 4214 | . . . 4 ⊢ ∅ ∈ V | |
| 6 | prssg 3828 | . . . 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 949 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽) |
| 9 | toponuni 14729 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
| 10 | eqimss2 3280 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝑋) | |
| 11 | 9, 10 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ⊆ 𝑋) |
| 12 | sspwuni 4053 | . . 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 1395 ∈ wcel 2200 Vcvv 2800 ⊆ wss 3198 ∅c0 3492 𝒫 cpw 3650 {cpr 3668 ∪ cuni 3891 ‘cfv 5324 Topctop 14711 TopOnctopon 14724 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-mpt 4150 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-top 14712 df-topon 14725 |
| This theorem is referenced by: (None) |
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