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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 44115 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
| 5 | difssd 4137 | . 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵) | |
| 6 | 4, 5 | sselpwd 5328 | 1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 𝒫 cpw 4600 class class class wbr 5143 ∘ ccom 5689 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: clsneiel2 44122 |
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