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Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgrcomplex | Structured version Visualization version GIF version |
Description: The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.) |
Ref | Expression |
---|---|
neicvgbex.d | ⊢ 𝐷 = (𝑃‘𝐵) |
neicvgbex.h | ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) |
neicvgbex.r | ⊢ (𝜑 → 𝑁𝐻𝑀) |
Ref | Expression |
---|---|
neicvgrcomplex | ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neicvgbex.d | . . 3 ⊢ 𝐷 = (𝑃‘𝐵) | |
2 | neicvgbex.h | . . 3 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
3 | neicvgbex.r | . . 3 ⊢ (𝜑 → 𝑁𝐻𝑀) | |
4 | 1, 2, 3 | neicvgbex 42848 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | difssd 4131 | . 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵) | |
6 | 4, 5 | sselpwd 5325 | 1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∖ cdif 3944 𝒫 cpw 4601 class class class wbr 5147 ∘ ccom 5679 ‘cfv 6540 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-iota 6492 df-fv 6548 |
This theorem is referenced by: neicvgel2 42856 |
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