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Mirrors > Home > MPE Home > Th. List > toponss | Structured version Visualization version GIF version |
Description: A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
toponss | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 4403 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ ∪ 𝐽) |
3 | toponuni 20542 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝑋 = ∪ 𝐽) |
5 | 2, 4 | sseqtr4d 3605 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 TopOnctopon 20518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-topon 20523 |
This theorem is referenced by: en2top 20600 neiptopreu 20747 iscnp3 20858 cnntr 20889 cncnp 20894 isreg2 20991 connsub 21034 iunconlem 21040 concompclo 21048 1stccnp 21075 kgenidm 21160 tx1cn 21222 tx2cn 21223 xkoccn 21232 txcnp 21233 ptcnplem 21234 xkoinjcn 21300 idqtop 21319 qtopss 21328 kqfvima 21343 kqsat 21344 kqreglem1 21354 kqreglem2 21355 qtopf1 21429 fbflim 21590 flimcf 21596 flimrest 21597 isflf 21607 fclscf 21639 subgntr 21720 ghmcnp 21728 qustgpopn 21733 qustgplem 21734 tsmsxplem1 21766 tsmsxp 21768 ressusp 21879 mopnss 22061 xrtgioo 22417 lebnumlem2 22569 cfilfcls 22880 iscmet3lem2 22898 dvres3a 23484 dvmptfsum 23542 dvcnvlem 23543 dvcnv 23544 efopn 24204 dvatan 24462 txomap 29229 cnllyscon 30481 cvmlift2lem9a 30539 icccncfext 38773 dvmptconst 38803 dvmptidg 38805 qndenserrnopnlem 39193 opnvonmbllem2 39523 |
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