Theorem 0he 44235

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 6031	. . 3 ⊢ (∅ “ 𝐴) = ∅
2		0ss 4329	. . 3 ⊢ ∅ ⊆ 𝐴
3	1, 2	eqsstri 3961	. 2 ⊢ (∅ “ 𝐴) ⊆ 𝐴
4		df-he 44226	. 2 ⊢ (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
5	3, 4	mpbir 232	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ⊆ wss 3883  ∅c0 4262   “ cima 5622   hereditary whe 44225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-he 44226

This theorem is referenced by: (None)