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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 40489 | . 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴 | |
2 | idhe 40488 | . 2 ⊢ I hereditary 𝒫 𝐴 | |
3 | unhe1 40486 | . 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 3879 𝒫 cpw 4497 I cid 5424 ◡ccnv 5518 [⊊] crpss 7428 hereditary whe 40473 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-rpss 7429 df-he 40474 |
This theorem is referenced by: (None) |
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