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Mirrors > Home > MPE Home > Th. List > t1sep2 | Structured version Visualization version GIF version |
Description: Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.) |
Ref | Expression |
---|---|
t1sep.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
t1sep2 | ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t1top 21642 | . . . . . 6 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | |
2 | t1sep.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | toptopon 21229 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
4 | 1, 3 | sylib 210 | . . . . 5 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋)) |
5 | ist1-2 21659 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝐽 ∈ Fre → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) |
7 | 6 | ibi 259 | . . 3 ⊢ (𝐽 ∈ Fre → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)) |
8 | eleq1 2854 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)) | |
9 | 8 | imbi1d 334 | . . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜))) |
10 | 9 | ralbidv 3148 | . . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜))) |
11 | eqeq1 2783 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑦 ↔ 𝐴 = 𝑦)) | |
12 | 10, 11 | imbi12d 337 | . . . 4 ⊢ (𝑥 = 𝐴 → ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝐴 = 𝑦))) |
13 | eleq1 2854 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)) | |
14 | 13 | imbi2d 333 | . . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜))) |
15 | 14 | ralbidv 3148 | . . . . 5 ⊢ (𝑦 = 𝐵 → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜))) |
16 | eqeq2 2790 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐴 = 𝐵)) | |
17 | 15, 16 | imbi12d 337 | . . . 4 ⊢ (𝑦 = 𝐵 → ((∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))) |
18 | 12, 17 | rspc2v 3549 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))) |
19 | 7, 18 | mpan9 499 | . 2 ⊢ ((𝐽 ∈ Fre ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵)) |
20 | 19 | 3impb 1095 | 1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → 𝐵 ∈ 𝑜) → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ∀wral 3089 ∪ cuni 4712 ‘cfv 6188 Topctop 21205 TopOnctopon 21222 Frect1 21619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-topgen 16573 df-top 21206 df-topon 21223 df-cld 21331 df-t1 21626 |
This theorem is referenced by: t1sep 21682 isr0 22049 |
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