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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 2991 | . . 3 ⊢ (x = A → x ⊆ A) | |
2 | elsn 3382 | . . 3 ⊢ (x ∈ {A} ↔ x = A) | |
3 | df-pw 3353 | . . . 4 ⊢ 𝒫 A = {x ∣ x ⊆ A} | |
4 | 3 | abeq2i 2145 | . . 3 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A) |
5 | 1, 2, 4 | 3imtr4i 190 | . 2 ⊢ (x ∈ {A} → x ∈ 𝒫 A) |
6 | 5 | ssriv 2943 | 1 ⊢ {A} ⊆ 𝒫 A |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 ⊆ wss 2911 𝒫 cpw 3351 {csn 3367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 |
This theorem is referenced by: snexgOLD 3926 snexg 3927 |
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