Theorem sshepw 44234

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 44233	. 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴
2		idhe 44232	. 2 ⊢ I hereditary 𝒫 𝐴
3		unhe1 44230	. 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴)
4	1, 2, 3	mp2an 693	1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴

Syntax hints:   ∪ cun 3888  𝒫 cpw 4542   I cid 5518  ^◡ccnv 5623   [⊊] crpss 7669   hereditary whe 44217

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-rpss 7670  df-he 44218

This theorem is referenced by: (None)