Theorem sshepw 43191

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

Step	Hyp	Ref	Expression
1		psshepw 43190	. 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴
2		idhe 43189	. 2 ⊢ I hereditary 𝒫 𝐴
3		unhe1 43187	. 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴)
4	1, 2, 3	mp2an 691	1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴

Syntax hints:   ∪ cun 3942  𝒫 cpw 4598   I cid 5569  ^◡ccnv 5671   [⊊] crpss 7721   hereditary whe 43174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-rpss 7722  df-he 43175

This theorem is referenced by: (None)