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Theorem psshepw 42587
Description: The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
psshepw [] hereditary 𝒫 𝐴

Proof of Theorem psshepw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfhe3 42574 . 2 ( [] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴)))
2 sstr2 3990 . . . . 5 (𝑦𝑥 → (𝑥𝐴𝑦𝐴))
3 pssss 4096 . . . . 5 (𝑦𝑥𝑦𝑥)
42, 3syl11 33 . . . 4 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
54alrimiv 1931 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐴))
6 velpw 4608 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 vex 3479 . . . . . . 7 𝑥 ∈ V
8 vex 3479 . . . . . . 7 𝑦 ∈ V
97, 8brcnv 5883 . . . . . 6 (𝑥 [] 𝑦𝑦 [] 𝑥)
107brrpss 7716 . . . . . 6 (𝑦 [] 𝑥𝑦𝑥)
119, 10bitri 275 . . . . 5 (𝑥 [] 𝑦𝑦𝑥)
12 velpw 4608 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1311, 12imbi12i 351 . . . 4 ((𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ (𝑦𝑥𝑦𝐴))
1413albii 1822 . . 3 (∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
155, 6, 143imtr4i 292 . 2 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴))
161, 15mpgbir 1802 1 [] hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wcel 2107  wss 3949  wpss 3950  𝒫 cpw 4603   class class class wbr 5149  ccnv 5676   [] crpss 7712   hereditary whe 42571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-rpss 7713  df-he 42572
This theorem is referenced by:  sshepw  42588
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