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Theorem clsneircomplex 42356
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 42355 . 2 (𝜑𝐵 ∈ V)
5 difssd 4092 . 2 (𝜑 → (𝐵𝑆) ⊆ 𝐵)
64, 5sselpwd 5283 1 (𝜑 → (𝐵𝑆) ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  𝒫 cpw 4560   class class class wbr 5105  ccom 5637  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-iota 6448  df-fv 6504
This theorem is referenced by:  clsneiel2  42362
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