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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 3524 | . . 3 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴}) | |
2 | 1 | ss2abi 3012 | . 2 ⊢ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥 ∣ 𝑥 ⊆ {𝐴}} |
3 | dfpr2 3394 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
4 | df-pw 3361 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
5 | 2, 3, 4 | 3sstr4i 2984 | 1 ⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 629 = wceq 1243 {cab 2026 ⊆ wss 2917 ∅c0 3224 𝒫 cpw 3359 {csn 3375 {cpr 3376 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 |
This theorem is referenced by: pwpw0ss 3575 |
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