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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 3603 | . . 3 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴}) | |
2 | 1 | ss2abi 3094 | . 2 ⊢ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥 ∣ 𝑥 ⊆ {𝐴}} |
3 | dfpr2 3469 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
4 | df-pw 3435 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
5 | 2, 3, 4 | 3sstr4i 3066 | 1 ⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 665 = wceq 1290 {cab 2075 ⊆ wss 3000 ∅c0 3287 𝒫 cpw 3433 {csn 3450 {cpr 3451 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 |
This theorem is referenced by: pwpw0ss 3654 |
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