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Theorem clsneircomplex 42467
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 42466 . 2 (𝜑𝐵 ∈ V)
5 difssd 4096 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
64, 5sselpwd 5287 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3447  cdif 3911  𝒫 cpw 4564   class class class wbr 5109  ccom 5641  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fv 6508
This theorem is referenced by:  clsneiel2  42473
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