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Mirrors > Home > MPE Home > Th. List > sst0 | Structured version Visualization version GIF version |
Description: A topology finer than a T0 topology is T0. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
t1sep.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
sst0 | ⊢ ((𝐽 ∈ Kol2 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ Kol2) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t1sep.1 | . 2 ⊢ 𝑋 = ∪ 𝐽 | |
2 | t0top 21920 | . 2 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) | |
3 | cnt0 21937 | . 2 ⊢ ((𝐽 ∈ Kol2 ∧ ( I ↾ 𝑋):𝑋–1-1→𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Kol2) | |
4 | 1, 2, 3 | sshauslem 21963 | 1 ⊢ ((𝐽 ∈ Kol2 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ Kol2) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3924 ∪ cuni 4824 I cid 5445 ↾ cres 5543 ‘cfv 6341 TopOnctopon 21501 Kol2ct0 21897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-map 8394 df-top 21485 df-topon 21502 df-cn 21818 df-t0 21904 |
This theorem is referenced by: (None) |
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