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Theorem clsneircomplex 38921
 Description: The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)
Hypotheses
Ref Expression
clsneibex.d 𝐷 = (𝑃𝐵)
clsneibex.h 𝐻 = (𝐹𝐷)
clsneibex.r (𝜑𝐾𝐻𝑁)
Assertion
Ref Expression
clsneircomplex (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)

Proof of Theorem clsneircomplex
StepHypRef Expression
1 clsneibex.d . . 3 𝐷 = (𝑃𝐵)
2 clsneibex.h . . 3 𝐻 = (𝐹𝐷)
3 clsneibex.r . . 3 (𝜑𝐾𝐻𝑁)
41, 2, 3clsneibex 38920 . 2 (𝜑𝐵 ∈ V)
5 difssd 3881 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
64, 5sselpwd 4959 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∖ cdif 3712  𝒫 cpw 4302   class class class wbr 4804   ∘ ccom 5270  ‘cfv 6049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fv 6057 This theorem is referenced by:  clsneiel2  38927
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