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Theorem 0he 41959
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 6029 . . 3 (∅ “ 𝐴) = ∅
2 0ss 4355 . . 3 ∅ ⊆ 𝐴
31, 2eqsstri 3977 . 2 (∅ “ 𝐴) ⊆ 𝐴
4 df-he 41950 . 2 (∅ hereditary 𝐴 ↔ (∅ “ 𝐴) ⊆ 𝐴)
53, 4mpbir 230 1 ∅ hereditary 𝐴
Colors of variables: wff setvar class
Syntax hints:  wss 3909  c0 4281  cima 5635   hereditary whe 41949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-he 41950
This theorem is referenced by: (None)
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