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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 38917 | . 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴 | |
2 | idhe 38916 | . 2 ⊢ I hereditary 𝒫 𝐴 | |
3 | unhe1 38914 | . 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴) | |
4 | 1, 2, 3 | mp2an 683 | 1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3796 𝒫 cpw 4380 I cid 5251 ◡ccnv 5345 [⊊] crpss 7201 hereditary whe 38901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-rpss 7202 df-he 38902 |
This theorem is referenced by: (None) |
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