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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 13405 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | 0opn 13397 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∅ ∈ 𝐽) |
4 | toponmax 13416 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) | |
5 | 0ex 4130 | . . . 4 ⊢ ∅ ∈ V | |
6 | prssg 3749 | . . . 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 943 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → {∅, 𝑋} ⊆ 𝐽) |
9 | toponuni 13406 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
10 | eqimss2 3210 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝑋) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 ⊆ 𝑋) |
12 | sspwuni 3971 | . . 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 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 ∅c0 3422 𝒫 cpw 3575 {cpr 3593 ∪ cuni 3809 ‘cfv 5216 Topctop 13388 TopOnctopon 13401 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-top 13389 df-topon 13402 |
This theorem is referenced by: (None) |
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