Theorem ntrclsrcomplex 45296

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 45295	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
4		difssd 4076	. 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵)
5	3, 4	sselpwd 5297	1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048  𝒫 cpw 4560   class class class wbr 5108  ‘cfv 6566

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574

This theorem is used by:  ntrclsfveq1  45321  ntrclsfveq2  45322  ntrclsfveq  45323  ntrclsss  45324  ntrclsneine0lem  45325  ntrclsk2  45329  ntrclskb  45330  ntrclsk4  45333