Theorem he0 43746

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 6106	. . 3 ⊢ (𝐴 “ ∅) = ∅
2	1	eqimssi 4069	. 2 ⊢ (𝐴 “ ∅) ⊆ ∅
3		df-he 43735	. 2 ⊢ (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
4	2, 3	mpbir 231	1 ⊢ 𝐴 hereditary ∅

Syntax hints:   ⊆ wss 3976  ∅c0 4352   “ cima 5703   hereditary whe 43734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-he 43735

This theorem is referenced by: (None)