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Theorem sshepw 40407
Description: The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
sshepw ( [] ∪ I ) hereditary 𝒫 𝐴

Proof of Theorem sshepw
StepHypRef Expression
1 psshepw 40406 . 2 [] hereditary 𝒫 𝐴
2 idhe 40405 . 2 I hereditary 𝒫 𝐴
3 unhe1 40403 . 2 (( [] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → ( [] ∪ I ) hereditary 𝒫 𝐴)
41, 2, 3mp2an 691 1 ( [] ∪ I ) hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  cun 3917  𝒫 cpw 4522   I cid 5446  ccnv 5541   [] crpss 7442   hereditary whe 40390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-rpss 7443  df-he 40391
This theorem is referenced by: (None)
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