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Mirrors > Home > MPE Home > Th. List > snsspw | Structured version Visualization version GIF version |
Description: The singleton of a class is a subset of its power class. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
snsspw | ⊢ {𝐴} ⊆ 𝒫 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 4023 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴) | |
2 | velsn 4583 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | velpw 4544 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
4 | 1, 2, 3 | 3imtr4i 294 | . 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴) |
5 | 4 | ssriv 3971 | 1 ⊢ {𝐴} ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-in 3943 df-ss 3952 df-pw 4541 df-sn 4568 |
This theorem is referenced by: snexALT 5284 snwf 9238 tsksn 10182 mnusnd 40624 |
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