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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 3688 | . . 3 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴}) | |
2 | 1 | ss2abi 3174 | . 2 ⊢ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥 ∣ 𝑥 ⊆ {𝐴}} |
3 | dfpr2 3551 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
4 | df-pw 3517 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
5 | 2, 3, 4 | 3sstr4i 3143 | 1 ⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 698 = wceq 1332 {cab 2126 ⊆ wss 3076 ∅c0 3368 𝒫 cpw 3515 {csn 3532 {cpr 3533 |
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 604 ax-in2 605 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-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 |
This theorem is referenced by: pwpw0ss 3739 |
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