Theorem he0 38913

Description: Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)

Assertion

Ref	Expression
he0	⊢ 𝐴 hereditary ∅

Proof of Theorem he0

Step	Hyp	Ref	Expression
1		ima0 5726	. . 3 ⊢ (𝐴 “ ∅) = ∅
2	1	eqimssi 3884	. 2 ⊢ (𝐴 “ ∅) ⊆ ∅
3		df-he 38902	. 2 ⊢ (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
4	2, 3	mpbir 223	1 ⊢ 𝐴 hereditary ∅

Syntax hints:   ⊆ wss 3798  ∅c0 4146   “ cima 5349   hereditary whe 38901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-he 38902

This theorem is referenced by: (None)