![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eltop3 | Structured version Visualization version GIF version |
Description: Membership in a topology. (Contributed by NM, 19-Jul-2006.) |
Ref | Expression |
---|---|
eltop3 | ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgtop 21185 | . . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
2 | 1 | eleq2d 2845 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (topGen‘𝐽) ↔ 𝐴 ∈ 𝐽)) |
3 | eltg3 21174 | . 2 ⊢ (𝐽 ∈ Top → (𝐴 ∈ (topGen‘𝐽) ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥))) | |
4 | 2, 3 | bitr3d 273 | 1 ⊢ (𝐽 ∈ Top → (𝐴 ∈ 𝐽 ↔ ∃𝑥(𝑥 ⊆ 𝐽 ∧ 𝐴 = ∪ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ⊆ wss 3792 ∪ cuni 4671 ‘cfv 6135 topGenctg 16484 Topctop 21105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-topgen 16490 df-top 21106 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |