Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 40445 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | difssd 4108 | . 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵) | |
6 | 4, 5 | sselpwd 5222 | 1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 𝒫 cpw 4538 class class class wbr 5058 ∘ ccom 5553 ‘cfv 6349 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fv 6357 |
This theorem is referenced by: clsneiel2 40452 |
Copyright terms: Public domain | W3C validator |