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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 43190 | . 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴 | |
2 | idhe 43189 | . 2 ⊢ I hereditary 𝒫 𝐴 | |
3 | unhe1 43187 | . 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3942 𝒫 cpw 4598 I cid 5569 ◡ccnv 5671 [⊊] crpss 7721 hereditary whe 43174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-rpss 7722 df-he 43175 |
This theorem is referenced by: (None) |
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