Theorem psshepw 38082

Description: The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)

Assertion

Ref	Expression
psshepw	|- `' [ C.] hereditary ~P A

Proof of Theorem psshepw

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfhe3 38069	. 2 |- ( `' [ C.] hereditary ~P A <-> A. x ( x e. ~P A -> A. y ( x `' [ C.] y -> y e. ~P A ) ) )
2		sstr2 3610	. . . . 5 |- ( y C_ x -> ( x C_ A -> y C_ A ) )
3		pssss 3702	. . . . 5 |- ( y C. x -> y C_ x )
4	2, 3	syl11 33	. . . 4 |- ( x C_ A -> ( y C. x -> y C_ A ) )
5	4	alrimiv 1855	. . 3 |- ( x C_ A -> A. y ( y C. x -> y C_ A ) )
6		selpw 4165	. . 3 |- ( x e. ~P A <-> x C_ A )
7		vex 3203	. . . . . . 7 |- x e. _V
8		vex 3203	. . . . . . 7 |- y e. _V
9	7, 8	brcnv 5305	. . . . . 6 |- ( x `' [ C.] y <-> y [ C.] x )
10	7	brrpss 6940	. . . . . 6 |- ( y [ C.] x <-> y C. x )
11	9, 10	bitri 264	. . . . 5 |- ( x `' [ C.] y <-> y C. x )
12		selpw 4165	. . . . 5 |- ( y e. ~P A <-> y C_ A )
13	11, 12	imbi12i 340	. . . 4 |- ( ( x `' [ C.] y -> y e. ~P A ) <-> ( y C. x -> y C_ A ) )
14	13	albii 1747	. . 3 |- ( A. y ( x `' [ C.] y -> y e. ~P A ) <-> A. y ( y C. x -> y C_ A ) )
15	5, 6, 14	3imtr4i 281	. 2 |- ( x e. ~P A -> A. y ( x `' [ C.] y -> y e. ~P A ) )
16	1, 15	mpgbir 1726	1 |- `' [ C.] hereditary ~P A

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   C_ wss 3574   C. wpss 3575   ~Pcpw 4158   class class class wbr 4653   `'ccnv 5113   [ C.] crpss 6936   hereditary whe 38066

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-rpss 6937  df-he 38067

This theorem is referenced by:  sshepw  38083