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Theorem he0 43214
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 6080 . . 3 (𝐴 “ ∅) = ∅
21eqimssi 4040 . 2 (𝐴 “ ∅) ⊆ ∅
3 df-he 43203 . 2 (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
42, 3mpbir 230 1 𝐴 hereditary ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3947  c0 4323  cima 5681   hereditary whe 43202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-he 43203
This theorem is referenced by: (None)
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