Theorem sshepw 38083

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

Assertion

Ref	Expression
sshepw	|- ( `' [ C.] u. _I ) hereditary ~P A

Proof of Theorem sshepw

Step	Hyp	Ref	Expression
1		psshepw 38082	. 2 |- `' [ C.] hereditary ~P A
2		idhe 38081	. 2 |- _I hereditary ~P A
3		unhe1 38079	. 2 |- ( ( `' [ C.] hereditary ~P A /\ _I hereditary ~P A ) -> ( `' [ C.] u. _I ) hereditary ~P A )
4	1, 2, 3	mp2an 708	1 |- ( `' [ C.] u. _I ) hereditary ~P A

Syntax hints:   u. cun 3572   ~Pcpw 4158   _I cid 5023   `'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-id 5024  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: (None)