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

Theorem he0 38913
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 5726 . . 3 (𝐴 “ ∅) = ∅
21eqimssi 3884 . 2 (𝐴 “ ∅) ⊆ ∅
3 df-he 38902 . 2 (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
42, 3mpbir 223 1 𝐴 hereditary ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3798  c0 4146  cima 5349   hereditary whe 38901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-xp 5352  df-cnv 5354  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-he 38902
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator