Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  he0 Structured version   Visualization version   GIF version

Theorem he0 40416
 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
StepHypRef Expression
1 ima0 5923 . . 3 (𝐴 “ ∅) = ∅
21eqimssi 4000 . 2 (𝐴 “ ∅) ⊆ ∅
3 df-he 40405 . 2 (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
42, 3mpbir 234 1 𝐴 hereditary ∅
 Colors of variables: wff setvar class Syntax hints:   ⊆ wss 3908  ∅c0 4265   “ cima 5535   hereditary whe 40404 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-he 40405 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator