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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 4859 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | 1 | adantl 482 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ ∪ 𝐽) |
3 | toponuni 21450 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
4 | 3 | adantr 481 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝑋 = ∪ 𝐽) |
5 | 2, 4 | sseqtrrd 4005 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ∪ cuni 4830 ‘cfv 6348 TopOnctopon 21446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-topon 21447 |
This theorem is referenced by: en2top 21521 neiptopreu 21669 iscnp3 21780 cnntr 21811 cncnp 21816 isreg2 21913 connsub 21957 iunconnlem 21963 conncompclo 21971 1stccnp 21998 kgenidm 22083 tx1cn 22145 tx2cn 22146 xkoccn 22155 txcnp 22156 ptcnplem 22157 xkoinjcn 22223 idqtop 22242 qtopss 22251 kqfvima 22266 kqsat 22267 kqreglem1 22277 kqreglem2 22278 qtopf1 22352 fbflim 22512 flimcf 22518 flimrest 22519 isflf 22529 fclscf 22561 subgntr 22642 ghmcnp 22650 qustgpopn 22655 qustgplem 22656 tsmsxplem1 22688 tsmsxp 22690 ressusp 22801 mopnss 22983 xrtgioo 23341 lebnumlem2 23493 cfilfcls 23804 iscmet3lem2 23822 dvres3a 24439 dvmptfsum 24499 dvcnvlem 24500 dvcnv 24501 efopn 25168 txomap 30997 cnllysconn 32389 cvmlift2lem9a 32447 icccncfext 42046 dvmptconst 42075 dvmptidg 42077 qndenserrnopnlem 42459 opnvonmbllem2 42792 |
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