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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 3909 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 ⊆ 𝐴) | |
2 | velsn 4451 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
3 | selpw 4423 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
4 | 1, 2, 3 | 3imtr4i 284 | . 2 ⊢ (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝒫 𝐴) |
5 | 4 | ssriv 3858 | 1 ⊢ {𝐴} ⊆ 𝒫 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2048 ⊆ wss 3825 𝒫 cpw 4416 {csn 4435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-v 3411 df-in 3832 df-ss 3839 df-pw 4418 df-sn 4436 |
This theorem is referenced by: snexALT 5130 snwf 9024 tsksn 9972 mnusnd 39924 |
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