Theorem ntrclsrcomplex 44192

Description: The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 25-Jun-2021.)

Hypotheses

Ref	Expression
ntrclsbex.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrclsbex.r	⊢ (𝜑 → 𝐼𝐷𝐾)

Assertion

Ref	Expression
ntrclsrcomplex	⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Proof of Theorem ntrclsrcomplex

Step	Hyp	Ref	Expression
1		ntrclsbex.d	. . 3 ⊢ 𝐷 = (𝑂‘𝐵)
2		ntrclsbex.r	. . 3 ⊢ (𝜑 → 𝐼𝐷𝐾)
3	1, 2	ntrclsbex 44191	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
4		difssd 4086	. 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵)
5	3, 4	sselpwd 5270	1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∖ cdif 3895  𝒫 cpw 4551   class class class wbr 5095  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497

This theorem is referenced by:  ntrclsfveq1  44217  ntrclsfveq2  44218  ntrclsfveq  44219  ntrclsss  44220  ntrclsneine0lem  44221  ntrclsk2  44225  ntrclskb  44226  ntrclsk4  44229