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Theorem sshepw 38609
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 38608 . 2 [] hereditary 𝒫 𝐴
2 idhe 38607 . 2 I hereditary 𝒫 𝐴
3 unhe1 38605 . 2 (( [] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → ( [] ∪ I ) hereditary 𝒫 𝐴)
41, 2, 3mp2an 672 1 ( [] ∪ I ) hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  cun 3721  𝒫 cpw 4297   I cid 5156  ccnv 5248   [] crpss 7083   hereditary whe 38592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-rpss 7084  df-he 38593
This theorem is referenced by: (None)
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