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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneircomplex | 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 |
---|---|
clsneibex.d | ⊢ 𝐷 = (𝑃‘𝐵) |
clsneibex.h | ⊢ 𝐻 = (𝐹 ∘ 𝐷) |
clsneibex.r | ⊢ (𝜑 → 𝐾𝐻𝑁) |
Ref | Expression |
---|---|
clsneircomplex | ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsneibex.d | . . 3 ⊢ 𝐷 = (𝑃‘𝐵) | |
2 | clsneibex.h | . . 3 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
3 | clsneibex.r | . . 3 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
4 | 1, 2, 3 | clsneibex 44092 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | difssd 4147 | . 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵) | |
6 | 4, 5 | sselpwd 5334 | 1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 𝒫 cpw 4605 class class class wbr 5148 ∘ ccom 5693 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fv 6571 |
This theorem is referenced by: clsneiel2 44099 |
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