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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sshepw | Structured version Visualization version GIF version | ||
| Description: The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.) |
| Ref | Expression |
|---|---|
| sshepw | ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | psshepw 43784 | . 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴 | |
| 2 | idhe 43783 | . 2 ⊢ I hereditary 𝒫 𝐴 | |
| 3 | unhe1 43781 | . 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∪ cun 3915 𝒫 cpw 4566 I cid 5535 ◡ccnv 5640 [⊊] crpss 7701 hereditary whe 43768 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2709 df-cleq 2722 df-clel 2804 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-br 5111 df-opab 5173 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-rpss 7702 df-he 43769 |
| This theorem is referenced by: (None) |
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