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Theorem psshepw 41726
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 41713 . 2 ( [] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴)))
2 sstr2 3939 . . . . 5 (𝑦𝑥 → (𝑥𝐴𝑦𝐴))
3 pssss 4042 . . . . 5 (𝑦𝑥𝑦𝑥)
42, 3syl11 33 . . . 4 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
54alrimiv 1929 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐴))
6 velpw 4552 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 vex 3445 . . . . . . 7 𝑥 ∈ V
8 vex 3445 . . . . . . 7 𝑦 ∈ V
97, 8brcnv 5824 . . . . . 6 (𝑥 [] 𝑦𝑦 [] 𝑥)
107brrpss 7641 . . . . . 6 (𝑦 [] 𝑥𝑦𝑥)
119, 10bitri 274 . . . . 5 (𝑥 [] 𝑦𝑦𝑥)
12 velpw 4552 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1311, 12imbi12i 350 . . . 4 ((𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ (𝑦𝑥𝑦𝐴))
1413albii 1820 . . 3 (∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
155, 6, 143imtr4i 291 . 2 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴))
161, 15mpgbir 1800 1 [] hereditary 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  wcel 2105  wss 3898  wpss 3899  𝒫 cpw 4547   class class class wbr 5092  ccnv 5619   [] crpss 7637   hereditary whe 41710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-rpss 7638  df-he 41711
This theorem is referenced by:  sshepw  41727
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