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Mirrors > Home > MPE Home > Th. List > ist1-5 | Structured version Visualization version GIF version |
Description: A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
ist1-5 | ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Fre)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t1t0 21955 | . 2 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2) | |
2 | t1hmph 22398 | . 2 ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre)) | |
3 | t1hmph 22398 | . 2 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Fre → 𝐽 ∈ Fre)) | |
4 | 1, 2, 3 | ist1-5lem 22427 | 1 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Fre)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ‘cfv 6354 Kol2ct0 21913 Frect1 21914 KQckq 22300 |
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-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-1o 8101 df-map 8407 df-topgen 16716 df-qtop 16779 df-top 21501 df-topon 21518 df-cld 21626 df-cn 21834 df-t0 21920 df-t1 21921 df-kq 22301 df-hmeo 22362 df-hmph 22363 |
This theorem is referenced by: (None) |
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