Theorem he0 43186

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 6074	. . 3 ⊢ (𝐴 “ ∅) = ∅
2	1	eqimssi 4038	. 2 ⊢ (𝐴 “ ∅) ⊆ ∅
3		df-he 43175	. 2 ⊢ (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
4	2, 3	mpbir 230	1 ⊢ 𝐴 hereditary ∅

Syntax hints:   ⊆ wss 3944  ∅c0 4318   “ cima 5675   hereditary whe 43174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-he 43175

This theorem is referenced by: (None)