Theorem ntrclsrcomplex 44391

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 44390	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
4		difssd 4091	. 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵)
5	3, 4	sselpwd 5275	1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900  𝒫 cpw 4556   class class class wbr 5100  ‘cfv 6500

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508

This theorem is referenced by:  ntrclsfveq1  44416  ntrclsfveq2  44417  ntrclsfveq  44418  ntrclsss  44419  ntrclsneine0lem  44420  ntrclsk2  44424  ntrclskb  44425  ntrclsk4  44428