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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 3830 | . . . . . . 7 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) | |
2 | 1 | unieqi 4381 | . . . . . 6 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) |
3 | unidif0 4764 | . . . . . 6 ⊢ ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 𝑥) | |
4 | 2, 3 | eqtri 2632 | . . . . 5 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥) |
5 | 4 | sseq2i 3593 | . . . 4 ⊢ (𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)) |
6 | 5 | abbii 2726 | . . 3 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} |
7 | difexg 4735 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V) | |
8 | tgval 20570 | . . . 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 20570 | . . 3 ⊢ (𝐵 ∈ V → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}) | |
11 | 6, 9, 10 | 3eqtr4a 2670 | . 2 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)) |
12 | difsnexi 6864 | . . . . 5 ⊢ ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V) | |
13 | 12 | con3i 149 | . . . 4 ⊢ (¬ 𝐵 ∈ V → ¬ (𝐵 ∖ {∅}) ∈ V) |
14 | fvprc 6097 | . . . 4 ⊢ (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅) |
16 | fvprc 6097 | . . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘𝐵) = ∅) | |
17 | 15, 16 | eqtr4d 2647 | . 2 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)) |
18 | 11, 17 | pm2.61i 175 | 1 ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 ‘cfv 5804 topGenctg 15921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-topgen 15927 |
This theorem is referenced by: prdsxmslem2 22144 |
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