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Theorem he0 38549
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 5627 . . 3 (𝐴 “ ∅) = ∅
21eqimssi 3788 . 2 (𝐴 “ ∅) ⊆ ∅
3 df-he 38538 . 2 (𝐴 hereditary ∅ ↔ (𝐴 “ ∅) ⊆ ∅)
42, 3mpbir 221 1 𝐴 hereditary ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3703  c0 4046  cima 5257   hereditary whe 38537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-he 38538
This theorem is referenced by: (None)
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