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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 41713 | . 2 ⊢ (◡ [⊊] hereditary 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → ∀𝑦(𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴))) | |
2 | sstr2 3939 | . . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ⊆ 𝐴 → 𝑦 ⊆ 𝐴)) | |
3 | pssss 4042 | . . . . 5 ⊢ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝑥) | |
4 | 2, 3 | syl11 33 | . . . 4 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
5 | 4 | alrimiv 1929 | . . 3 ⊢ (𝑥 ⊆ 𝐴 → ∀𝑦(𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
6 | velpw 4552 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
7 | vex 3445 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
8 | vex 3445 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
9 | 7, 8 | brcnv 5824 | . . . . . 6 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 [⊊] 𝑥) |
10 | 7 | brrpss 7641 | . . . . . 6 ⊢ (𝑦 [⊊] 𝑥 ↔ 𝑦 ⊊ 𝑥) |
11 | 9, 10 | bitri 274 | . . . . 5 ⊢ (𝑥◡ [⊊] 𝑦 ↔ 𝑦 ⊊ 𝑥) |
12 | velpw 4552 | . . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
13 | 11, 12 | imbi12i 350 | . . . 4 ⊢ ((𝑥◡ [⊊] 𝑦 → 𝑦 ∈ 𝒫 𝐴) ↔ (𝑦 ⊊ 𝑥 → 𝑦 ⊆ 𝐴)) |
14 | 13 | albii 1820 | . . 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 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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