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Mirrors > Home > ILE Home > Th. List > velpw | GIF version |
Description: Setvar variable membership in a power class (common case). See elpw 3516. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
velpw | ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . 2 ⊢ 𝑥 ∈ V | |
2 | 1 | elpw 3516 | 1 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 ⊆ wss 3071 𝒫 cpw 3510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-in 3077 df-ss 3084 df-pw 3512 |
This theorem is referenced by: ordpwsucss 4482 fabexg 5310 abexssex 6023 qsss 6488 mapval2 6572 pmsspw 6577 uniixp 6615 exmidpw 6802 fival 6858 npsspw 7282 restsspw 12133 istopon 12183 isbasis2g 12215 tgval2 12223 unitg 12234 distop 12257 cldss2 12278 ntreq0 12304 discld 12308 neisspw 12320 restdis 12356 cnntr 12397 |
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