Theorem he0 37098

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 5400	. . 3 ⊢ (𝐴 “ ∅) = ∅
2	1	eqimssi 3622	. 2 ⊢ (𝐴 “ ∅) ⊆ ∅
3		df-he 37087	. 2 ⊢ (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
4	2, 3	mpbir 220	1 ⊢ 𝐴 hereditary ∅

Syntax hints:   ⊆ wss 3540  ∅c0 3874   “ cima 5041   hereditary whe 37086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-he 37087

This theorem is referenced by: (None)