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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 4267 | . . . . . . 7 ⊢ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) | |
2 | 1 | unieqi 4915 | . . . . . 6 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) |
3 | unidif0 5354 | . . . . . 6 ⊢ ∪ ((𝐵 ∩ 𝒫 𝑥) ∖ {∅}) = ∪ (𝐵 ∩ 𝒫 𝑥) | |
4 | 2, 3 | eqtri 2755 | . . . . 5 ⊢ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) = ∪ (𝐵 ∩ 𝒫 𝑥) |
5 | 4 | sseq2i 4007 | . . . 4 ⊢ (𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥) ↔ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)) |
6 | 5 | abbii 2797 | . . 3 ⊢ {𝑥 ∣ 𝑥 ⊆ ∪ ((𝐵 ∖ {∅}) ∩ 𝒫 𝑥)} = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)} |
7 | difexg 5323 | . . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∖ {∅}) ∈ V) | |
8 | tgval 22851 | . . . 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 22851 | . . 3 ⊢ (𝐵 ∈ V → (topGen‘𝐵) = {𝑥 ∣ 𝑥 ⊆ ∪ (𝐵 ∩ 𝒫 𝑥)}) | |
11 | 6, 9, 10 | 3eqtr4a 2793 | . 2 ⊢ (𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)) |
12 | difsnexi 7757 | . . . 4 ⊢ ((𝐵 ∖ {∅}) ∈ V → 𝐵 ∈ V) | |
13 | fvprc 6883 | . . . 4 ⊢ (¬ (𝐵 ∖ {∅}) ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅) | |
14 | 12, 13 | nsyl5 159 | . . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = ∅) |
15 | fvprc 6883 | . . 3 ⊢ (¬ 𝐵 ∈ V → (topGen‘𝐵) = ∅) | |
16 | 14, 15 | eqtr4d 2770 | . 2 ⊢ (¬ 𝐵 ∈ V → (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵)) |
17 | 11, 16 | pm2.61i 182 | 1 ⊢ (topGen‘(𝐵 ∖ {∅})) = (topGen‘𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 {cab 2704 Vcvv 3469 ∖ cdif 3941 ∩ cin 3943 ⊆ wss 3944 ∅c0 4318 𝒫 cpw 4598 {csn 4624 ∪ cuni 4903 ‘cfv 6542 topGenctg 17412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-topgen 17418 |
This theorem is referenced by: prdsxmslem2 24431 |
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