Theorem ntrneircomplex 38372

Description: The relative complement of the class S exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)

Hypotheses

Ref	Expression
ntrnei.o	|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
ntrnei.f	|- F = ( ~P B O B )
ntrnei.r	|- ( ph -> I F N )

Assertion

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

Distinct variable groups:   i, j, k   i, l, j   i, m, j

Allowed substitution hints:   ph( i, j, k, m, l)   B( i, j, k, m, l)   S( i, j, k, m, l)   F( i, j, k, m, l)   I( i, j, k, m, l)   N( i, j, k, m, l)   O( i, j, k, m, l)

Proof of Theorem ntrneircomplex

Step	Hyp	Ref	Expression
1		ntrnei.o	. . 3 |- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
2		ntrnei.f	. . 3 |- F = ( ~P B O B )
3		ntrnei.r	. . 3 |- ( ph -> I F N )
4	1, 2, 3	ntrneibex 38371	. 2 |- ( ph -> B e. _V )
5		difssd 3738	. 2 |- ( ph -> ( B \ S ) C_ B )
6	4, 5	sselpwd 4807	1 |- ( ph -> ( B \ S ) e. ~P B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857

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  ax-pr 4906

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-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by: (None)