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Theorem he0 37898
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 5469 . . 3 (𝐴 “ ∅) = ∅
21eqimssi 3651 . 2 (𝐴 “ ∅) ⊆ ∅
3 df-he 37887 . 2 (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
42, 3mpbir 221 1 𝐴 hereditary ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3567  c0 3907  cima 5107   hereditary whe 37886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-xp 5110  df-cnv 5112  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-he 37887
This theorem is referenced by: (None)
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