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Theorem 0he 43795
Description: The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
Assertion
Ref Expression
0he ∅ hereditary 𝐴

Proof of Theorem 0he
StepHypRef Expression
1 0ima 6096 . . 3 (∅ “ 𝐴) = ∅
2 0ss 4400 . . 3 ∅ ⊆ 𝐴
31, 2eqsstri 4030 . 2 (∅ “ 𝐴) ⊆ 𝐴
4 df-he 43786 . 2 (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
53, 4mpbir 231 1 ∅ hereditary 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3951  c0 4333  cima 5688   hereditary whe 43785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-he 43786
This theorem is referenced by: (None)
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