Theorem ntrclsrcomplex 45144

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 45143	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
4		difssd 4072	. 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵)
5	3, 4	sselpwd 5290	1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∖ cdif 4044  𝒫 cpw 4556   class class class wbr 5102  ‘cfv 6543

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551

This theorem is used by:  ntrclsfveq1  45169  ntrclsfveq2  45170  ntrclsfveq  45171  ntrclsss  45172  ntrclsneine0lem  45173  ntrclsk2  45177  ntrclskb  45178  ntrclsk4  45181