Theorem 0he 38076

Description: The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)

Assertion

Ref	Expression
0he	|- (/) hereditary A

Proof of Theorem 0he

Step	Hyp	Ref	Expression
1		0ima 5482	. . 3 |- ( (/) " A ) = (/)
2		0ss 3972	. . 3 |- (/) C_ A
3	1, 2	eqsstri 3635	. 2 |- ( (/) " A ) C_ A
4		df-he 38067	. 2 |- ( (/) hereditary A <-> ( (/) " A ) C_ A )
5	3, 4	mpbir 221	1 |- (/) hereditary A

Syntax hints:   C_ wss 3574   (/)c0 3915   "cima 5117   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-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-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by: (None)