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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 37430 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | difssd 3700 | . 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵) | |
6 | 4, 5 | sselpwd 4734 | 1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 𝒫 cpw 4108 class class class wbr 4583 ∘ ccom 5042 ‘cfv 5804 |
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 |
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-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-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fv 5812 |
This theorem is referenced by: neicvgel2 37438 |
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