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Mirrors > Home > ILE Home > Th. List > pwidg | GIF version |
Description: Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
pwidg | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3047 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | elpwg 3443 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴)) | |
3 | 1, 2 | mpbiri 167 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 ⊆ wss 3002 𝒫 cpw 3435 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2624 df-in 3008 df-ss 3015 df-pw 3437 |
This theorem is referenced by: pwid 3450 axpweq 4014 baspartn 11811 epttop 11853 isopn3 11888 |
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