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| Mirrors > Home > MPE Home > Th. List > eltop2 | Structured version Visualization version GIF version | ||
| Description: Membership in a topology. (Contributed by NM, 19-Jul-2006.) | 
| Ref | Expression | 
|---|---|
| eltop2 | ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tgtop 22981 | . . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
| 2 | 1 | eleq2d 2826 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (topGen‘𝐽) ↔ 𝐴 ∈ 𝐽)) | 
| 3 | eltg2b 22967 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (topGen‘𝐽) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | |
| 4 | 2, 3 | bitr3d 281 | 1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 ‘cfv 6560 topGenctg 17483 Topctop 22900 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-topgen 17489 df-top 22901 | 
| This theorem is referenced by: isclo 23096 cncnp 23289 ist1-2 23356 hauscmp 23416 llycmpkgen2 23559 ptpjopn 23621 txkgen 23661 xkococn 23669 xkoinjcn 23696 fclscf 24034 subgntr 24116 opnsubg 24117 qustgpopn 24129 prdsxmslem2 24543 zdis 24839 efopn 26701 cvmopnlem 35284 neibastop3 36364 ioorrnopn 46325 ioorrnopnxr 46327 | 
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