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Theorem psshepw 37102
 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 37089 . 2 ( [] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴)))
2 pssss 3664 . . . . . 6 (𝑦𝑥𝑦𝑥)
3 sstr2 3575 . . . . . 6 (𝑦𝑥 → (𝑥𝐴𝑦𝐴))
42, 3syl 17 . . . . 5 (𝑦𝑥 → (𝑥𝐴𝑦𝐴))
54com12 32 . . . 4 (𝑥𝐴 → (𝑦𝑥𝑦𝐴))
65alrimiv 1842 . . 3 (𝑥𝐴 → ∀𝑦(𝑦𝑥𝑦𝐴))
7 selpw 4115 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
8 vex 3176 . . . . . . 7 𝑥 ∈ V
9 vex 3176 . . . . . . 7 𝑦 ∈ V
108, 9brcnv 5227 . . . . . 6 (𝑥 [] 𝑦𝑦 [] 𝑥)
118brrpss 6838 . . . . . 6 (𝑦 [] 𝑥𝑦𝑥)
1210, 11bitri 263 . . . . 5 (𝑥 [] 𝑦𝑦𝑥)
13 selpw 4115 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1412, 13imbi12i 339 . . . 4 ((𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ (𝑦𝑥𝑦𝐴))
1514albii 1737 . . 3 (∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
166, 7, 153imtr4i 280 . 2 (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥 [] 𝑦𝑦 ∈ 𝒫 𝐴))
171, 16mpgbir 1717 1 [] hereditary 𝒫 𝐴
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1473   ∈ wcel 1977   ⊆ wss 3540   ⊊ wpss 3541  𝒫 cpw 4108   class class class wbr 4583  ◡ccnv 5037   [⊊] crpss 6834   hereditary whe 37086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-rpss 6835  df-he 37087 This theorem is referenced by:  sshepw  37103
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