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Mirrors > Home > MPE Home > Th. List > Mathboxes > psshepw | Structured version Visualization version GIF version |
Description: The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
Ref | Expression |
---|---|
psshepw | ⊢ ◡ [⊊] hereditary 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfhe3 42828 | . 2 ⊢ (◡ [⊊] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴))) | |
2 | sstr2 3989 | . . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑦 ⊆ 𝐴)) | |
3 | pssss 4095 | . . . . 5 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥) | |
4 | 2, 3 | syl11 33 | . . . 4 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
5 | 4 | alrimiv 1930 | . . 3 ⊢ (𝑥 ⊆ 𝐴 → ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
6 | velpw 4607 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
7 | vex 3478 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | vex 3478 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 5882 | . . . . . 6 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 [⊊] 𝑥) |
10 | 7 | brrpss 7718 | . . . . . 6 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥) |
11 | 9, 10 | bitri 274 | . . . . 5 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 ⊊ 𝑥) |
12 | velpw 4607 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
13 | 11, 12 | imbi12i 350 | . . . 4 ⊢ ((𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
14 | 13 | albii 1821 | . . 3 ⊢ (∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
15 | 5, 6, 14 | 3imtr4i 291 | . 2 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴)) |
16 | 1, 15 | mpgbir 1801 | 1 ⊢ ◡ [⊊] hereditary 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∈ wcel 2106 ⊆ wss 3948 ⊊ wpss 3949 𝒫 cpw 4602 class class class wbr 5148 ◡ccnv 5675 [⊊] crpss 7714 hereditary whe 42825 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-11 2154 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-rpss 7715 df-he 42826 |
This theorem is referenced by: sshepw 42842 |
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