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Mirrors > Home > ILE Home > Th. List > snsspw | GIF version |
Description: The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
snsspw | ⊢ {𝐴} ⊆ 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 3151 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴) | |
2 | velsn 3544 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | df-pw 3512 | . . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
4 | 3 | abeq2i 2250 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
5 | 1, 2, 4 | 3imtr4i 200 | . 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴) |
6 | 5 | ssriv 3101 | 1 ⊢ {𝐴} ⊆ 𝒫 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 ⊆ wss 3071 𝒫 cpw 3510 {csn 3527 |
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 df-sn 3533 |
This theorem is referenced by: snexg 4108 |
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