Theorem 0he 39032

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 5736	. . 3 ⊢ (∅ “ 𝐴) = ∅
2		0ss 4198	. . 3 ⊢ ∅ ⊆ 𝐴
3	1, 2	eqsstri 3854	. 2 ⊢ (∅ “ 𝐴) ⊆ 𝐴
4		df-he 39023	. 2 ⊢ (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
5	3, 4	mpbir 223	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ⊆ wss 3792  ∅c0 4141   “ cima 5358   hereditary whe 39022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-xp 5361  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-he 39023

This theorem is referenced by: (None)