Theorem sshepw 43764

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 43763	. 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴
2		idhe 43762	. 2 ⊢ I hereditary 𝒫 𝐴
3		unhe1 43760	. 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴)
4	1, 2, 3	mp2an 692	1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴

Syntax hints:   ∪ cun 3929  𝒫 cpw 4580   I cid 5557  ^◡ccnv 5664   [⊊] crpss 7724   hereditary whe 43747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-rpss 7725  df-he 43748

This theorem is referenced by: (None)