Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > bastg | GIF version |
Description: A member of a basis is a subset of the topology it generates. (Contributed by NM, 16-Jul-2006.) (Revised by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
bastg | ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
2 | vex 2684 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
3 | 2 | pwid 3520 | . . . . . . 7 ⊢ 𝑥 ∈ 𝒫 𝑥 |
4 | 3 | a1i 9 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝒫 𝑥) |
5 | 1, 4 | elind 3256 | . . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (𝐵 ∩ 𝒫 𝑥)) |
6 | elssuni 3759 | . . . . 5 ⊢ (𝑥 ∈ (𝐵 ∩ 𝒫 𝑥) → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)) |
8 | 7 | ex 114 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐵 → 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) |
9 | eltg 12210 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ (topGen‘𝐵) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥))) | |
10 | 8, 9 | sylibrd 168 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐵 → 𝑥 ∈ (topGen‘𝐵))) |
11 | 10 | ssrdv 3098 | 1 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ⊆ (topGen‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ∩ cin 3065 ⊆ wss 3066 𝒫 cpw 3505 ∪ cuni 3731 ‘cfv 5118 topGenctg 12124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-topgen 12130 |
This theorem is referenced by: unitg 12220 tgclb 12223 tgtop 12226 tgidm 12232 tgss3 12236 bastop2 12242 tgcn 12366 tgcnp 12367 txopn 12423 txbasval 12425 blssopn 12643 xmettxlem 12667 iooretopg 12686 |
Copyright terms: Public domain | W3C validator |