![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tgdif0 | Structured version Visualization version GIF version |
Description: A generated topology is not affected by the addition or removal of the empty set from the base. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
tgdif0 | ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indif1 4198 | . . . . . . 7 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) | |
2 | 1 | unieqi 4813 | . . . . . 6 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) |
3 | unidif0 5225 | . . . . . 6 ⊢ ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 𝑥) | |
4 | 2, 3 | eqtri 2821 | . . . . 5 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥) |
5 | 4 | sseq2i 3944 | . . . 4 ⊢ (𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)) |
6 | 5 | abbii 2863 | . . 3 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} |
7 | difexg 5195 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V) | |
8 | tgval 21560 | . . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)}) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)}) |
10 | tgval 21560 | . . 3 ⊢ (𝐵 ∈ V → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}) | |
11 | 6, 9, 10 | 3eqtr4a 2859 | . 2 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)) |
12 | difsnexi 7463 | . . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V) | |
13 | fvprc 6638 | . . . 4 ⊢ (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅) | |
14 | 12, 13 | nsyl5 162 | . . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅) |
15 | fvprc 6638 | . . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘𝐵) = ∅) | |
16 | 14, 15 | eqtr4d 2836 | . 2 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)) |
17 | 11, 16 | pm2.61i 185 | 1 ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 {cab 2776 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 ∪ cuni 4800 ‘cfv 6324 topGenctg 16703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-topgen 16709 |
This theorem is referenced by: prdsxmslem2 23136 |
Copyright terms: Public domain | W3C validator |