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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 3574 | . . 3 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴}) | |
2 | 1 | ss2abi 3079 | . 2 ⊢ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥 ∣ 𝑥 ⊆ {𝐴}} |
3 | dfpr2 3444 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
4 | df-pw 3411 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
5 | 2, 3, 4 | 3sstr4i 3051 | 1 ⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 662 = wceq 1287 {cab 2071 ⊆ wss 2986 ∅c0 3272 𝒫 cpw 3409 {csn 3425 {cpr 3426 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-v 2616 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 |
This theorem is referenced by: pwpw0ss 3625 |
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