Theorem ntrclsrcomplex 38333

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	|- ( ph -> I D K )

Assertion

Ref	Expression
ntrclsrcomplex	|- ( ph -> ( B \ S ) e. ~P B )

Proof of Theorem ntrclsrcomplex

Step	Hyp	Ref	Expression
1		ntrclsbex.d	. . 3 |- D = ( O ` B )
2		ntrclsbex.r	. . 3 |- ( ph -> I D K )
3	1, 2	ntrclsbex 38332	. 2 |- ( ph -> B e. _V )
4		difssd 3738	. 2 |- ( ph -> ( B \ S ) C_ B )
5	3, 4	sselpwd 4807	1 |- ( ph -> ( B \ S ) e. ~P B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   class class class wbr 4653   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  ntrclsfveq1  38358  ntrclsfveq2  38359  ntrclsfveq  38360  ntrclsss  38361  ntrclsneine0lem  38362  ntrclsk2  38366  ntrclskb  38367  ntrclsk4  38370