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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 3608. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
velpw | ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2763 | . 2 ⊢ 𝑥 ∈ V | |
2 | 1 | elpw 3608 | 1 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2164 ⊆ wss 3154 𝒫 cpw 3602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-v 2762 df-in 3160 df-ss 3167 df-pw 3604 |
This theorem is referenced by: ordpwsucss 4600 fabexg 5442 abexssex 6179 qsss 6650 mapval2 6734 pmsspw 6739 uniixp 6777 exmidpw 6966 exmidpweq 6967 pw1fin 6968 pw1dc0el 6969 fival 7031 npsspw 7533 restsspw 12863 subsubrng2 13714 subsubrg2 13745 lssintclm 13883 istopon 14192 isbasis2g 14224 tgval2 14230 unitg 14241 distop 14264 cldss2 14285 ntreq0 14311 discld 14315 neisspw 14327 restdis 14363 cnntr 14404 |
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