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| Mirrors > Home > MPE Home > Th. List > pwsn | Structured version Visualization version GIF version | ||
| Description: The power set of a singleton. (Contributed by NM, 5-Jun-2006.) |
| Ref | Expression |
|---|---|
| pwsn | ⊢ 𝒫 {𝐴} = {∅, {𝐴}} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sssn 4826 | . . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) | |
| 2 | 1 | abbii 2809 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} |
| 3 | df-pw 4602 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
| 4 | dfpr2 4646 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
| 5 | 2, 3, 4 | 3eqtr4i 2775 | 1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}} |
| Colors of variables: wff setvar class |
| Syntax hints: ∨ wo 848 = wceq 1540 {cab 2714 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-pw 4602 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: pmtrsn 19537 topsn 22937 conncompid 23439 lfuhgr1v0e 29271 esumsnf 34065 cvmlift2lem9 35316 rrxtopn0b 46311 sge0sn 46394 |
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