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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 21724 | . . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
2 | 1 | eleq2d 2818 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (topGen‘𝐽) ↔ 𝐴 ∈ 𝐽)) |
3 | eltg2b 21710 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (topGen‘𝐽) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) | |
4 | 2, 3 | bitr3d 284 | 1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 ⊆ wss 3843 ‘cfv 6339 topGenctg 16814 Topctop 21644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-topgen 16820 df-top 21645 |
This theorem is referenced by: isclo 21838 cncnp 22031 ist1-2 22098 hauscmp 22158 llycmpkgen2 22301 ptpjopn 22363 txkgen 22403 xkococn 22411 xkoinjcn 22438 fclscf 22776 subgntr 22858 opnsubg 22859 qustgpopn 22871 prdsxmslem2 23282 zdis 23568 efopn 25401 cvmopnlem 32811 neibastop3 34194 ioorrnopn 43408 ioorrnopnxr 43410 |
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