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 4248 | . . . . . . 7 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) | |
2 | 1 | unieqi 4851 | . . . . . 6 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) |
3 | unidif0 5260 | . . . . . 6 ⊢ ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 𝑥) | |
4 | 2, 3 | eqtri 2844 | . . . . 5 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥) |
5 | 4 | sseq2i 3996 | . . . 4 ⊢ (𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)) |
6 | 5 | abbii 2886 | . . 3 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} |
7 | difexg 5231 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V) | |
8 | tgval 21563 | . . . 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 21563 | . . 3 ⊢ (𝐵 ∈ V → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}) | |
11 | 6, 9, 10 | 3eqtr4a 2882 | . 2 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)) |
12 | difsnexi 7483 | . . . . 5 ⊢ ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V) | |
13 | 12 | con3i 157 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ¬ (𝐵 ∖ {∅}) ∈ V) |
14 | fvprc 6663 | . . . 4 ⊢ (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅) |
16 | fvprc 6663 | . . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘𝐵) = ∅) | |
17 | 15, 16 | eqtr4d 2859 | . 2 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)) |
18 | 11, 17 | pm2.61i 184 | 1 ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 {cab 2799 Vcvv 3494 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ‘cfv 6355 topGenctg 16711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-topgen 16717 |
This theorem is referenced by: prdsxmslem2 23139 |
Copyright terms: Public domain | W3C validator |