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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 3156 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴) | |
2 | velsn 3549 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | df-pw 3517 | . . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴} | |
4 | 3 | abeq2i 2251 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) |
5 | 1, 2, 4 | 3imtr4i 200 | . 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴) |
6 | 5 | ssriv 3106 | 1 ⊢ {𝐴} ⊆ 𝒫 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 ⊆ wss 3076 𝒫 cpw 3515 {csn 3532 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 |
This theorem is referenced by: snexg 4116 |
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