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Mirrors > Home > ILE Home > Th. List > pwsnss | GIF version |
Description: The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.) |
Ref | Expression |
---|---|
pwsnss | ⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssnr 3680 | . . 3 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴}) | |
2 | 1 | ss2abi 3169 | . 2 ⊢ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥 ∣ 𝑥 ⊆ {𝐴}} |
3 | dfpr2 3546 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
4 | df-pw 3512 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
5 | 2, 3, 4 | 3sstr4i 3138 | 1 ⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 697 = wceq 1331 {cab 2125 ⊆ wss 3071 ∅c0 3363 𝒫 cpw 3510 {csn 3527 {cpr 3528 |
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-in1 603 ax-in2 604 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-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 |
This theorem is referenced by: pwpw0ss 3731 |
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