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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 40012 | . 2 ⊢ ◡ [⊊] hereditary 𝒫 𝐴 | |
2 | idhe 40011 | . 2 ⊢ I hereditary 𝒫 𝐴 | |
3 | unhe1 40009 | . 2 ⊢ ((◡ [⊊] hereditary 𝒫 𝐴 ∧ I hereditary 𝒫 𝐴) → (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3931 𝒫 cpw 4535 I cid 5452 ◡ccnv 5547 [⊊] crpss 7437 hereditary whe 39996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-rpss 7438 df-he 39997 |
This theorem is referenced by: (None) |
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