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Mirrors > Home > MPE Home > Th. List > connsub | Structured version Visualization version GIF version |
Description: Two equivalent ways of saying that a subspace topology is connected. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
connsub | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝑆)) → ¬ 𝑆 ⊆ (𝑥 ∪ 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | connsuba 22022 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝑆) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝑆) ≠ 𝑆))) | |
2 | inss1 4204 | . . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥 | |
3 | toponss 21529 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋) | |
4 | 3 | ad2ant2r 745 | . . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → 𝑥 ⊆ 𝑋) |
5 | 2, 4 | sstrid 3977 | . . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ 𝑋) |
6 | reldisj 4401 | . . . . . 6 ⊢ ((𝑥 ∩ 𝑦) ⊆ 𝑋 → (((𝑥 ∩ 𝑦) ∩ 𝑆) = ∅ ↔ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝑆))) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (((𝑥 ∩ 𝑦) ∩ 𝑆) = ∅ ↔ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝑆))) |
8 | 7 | 3anbi3d 1438 | . . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝑆) = ∅) ↔ ((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝑆)))) |
9 | sseqin2 4191 | . . . . . . 7 ⊢ (𝑆 ⊆ (𝑥 ∪ 𝑦) ↔ ((𝑥 ∪ 𝑦) ∩ 𝑆) = 𝑆) | |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (𝑆 ⊆ (𝑥 ∪ 𝑦) ↔ ((𝑥 ∪ 𝑦) ∩ 𝑆) = 𝑆)) |
11 | 10 | bicomd 225 | . . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (((𝑥 ∪ 𝑦) ∩ 𝑆) = 𝑆 ↔ 𝑆 ⊆ (𝑥 ∪ 𝑦))) |
12 | 11 | necon3abid 3052 | . . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → (((𝑥 ∪ 𝑦) ∩ 𝑆) ≠ 𝑆 ↔ ¬ 𝑆 ⊆ (𝑥 ∪ 𝑦))) |
13 | 8, 12 | imbi12d 347 | . . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐽)) → ((((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝑆) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝑆) ≠ 𝑆) ↔ (((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝑆)) → ¬ 𝑆 ⊆ (𝑥 ∪ 𝑦)))) |
14 | 13 | 2ralbidva 3198 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ ((𝑥 ∩ 𝑦) ∩ 𝑆) = ∅) → ((𝑥 ∪ 𝑦) ∩ 𝑆) ≠ 𝑆) ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝑆)) → ¬ 𝑆 ⊆ (𝑥 ∪ 𝑦)))) |
15 | 1, 14 | bitrd 281 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Conn ↔ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (((𝑥 ∩ 𝑆) ≠ ∅ ∧ (𝑦 ∩ 𝑆) ≠ ∅ ∧ (𝑥 ∩ 𝑦) ⊆ (𝑋 ∖ 𝑆)) → ¬ 𝑆 ⊆ (𝑥 ∪ 𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 ‘cfv 6349 (class class class)co 7150 ↾t crest 16688 TopOnctopon 21512 Conncconn 22013 |
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 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 df-er 8283 df-en 8504 df-fin 8507 df-fi 8869 df-rest 16690 df-topgen 16711 df-top 21496 df-topon 21513 df-bases 21548 df-cld 21621 df-conn 22014 |
This theorem is referenced by: iunconn 22030 clsconn 22032 reconn 23430 iunconnlem2 41262 |
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