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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 21580 | . . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
2 | 1 | eleq2d 2898 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (topGen‘𝐽) ↔ 𝐴 ∈ 𝐽)) |
3 | eltg2b 21566 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (topGen‘𝐽) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | |
4 | 2, 3 | bitr3d 283 | 1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ⊆ wss 3935 ‘cfv 6354 topGenctg 16710 Topctop 21500 |
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-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-topgen 16716 df-top 21501 |
This theorem is referenced by: isclo 21694 cncnp 21887 ist1-2 21954 hauscmp 22014 llycmpkgen2 22157 ptpjopn 22219 txkgen 22259 xkococn 22267 xkoinjcn 22294 fclscf 22632 subgntr 22714 opnsubg 22715 qustgpopn 22727 prdsxmslem2 23138 zdis 23423 efopn 25240 cvmopnlem 32525 neibastop3 33710 ioorrnopn 42589 ioorrnopnxr 42591 |
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