Theorem 0he 44359

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 6068	. . 3 ⊢ (∅ “ 𝐴) = ∅
2		0ss 4355	. . 3 ⊢ ∅ ⊆ 𝐴
3	1, 2	eqsstri 3983	. 2 ⊢ (∅ “ 𝐴) ⊆ 𝐴
4		df-he 44350	. 2 ⊢ (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
5	3, 4	mpbir 233	1 ⊢ ∅ hereditary 𝐴

Syntax hints:   ⊆ wss 3905  ∅c0 4286   “ cima 5651   hereditary whe 44349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-he 44350

This theorem is referenced by: (None)