Theorem he0 44462

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 6083	. . 3 ⊢ (𝐴 “ ∅) = ∅
2	1	eqimssi 4005	. 2 ⊢ (𝐴 “ ∅) ⊆ ∅
3		df-he 44451	. 2 ⊢ (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
4	2, 3	mpbir 234	1 ⊢ 𝐴 hereditary ∅

Syntax hints:   ⊆ wss 3913  ∅c0 4294   “ cima 5668   hereditary whe 44450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-he 44451

This theorem is referenced by: (None)