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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 4831 | . . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) | |
2 | 1 | abbii 2807 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} |
3 | df-pw 4607 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
4 | dfpr2 4651 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
5 | 2, 3, 4 | 3eqtr4i 2773 | 1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 847 = wceq 1537 {cab 2712 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 {cpr 4633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-pw 4607 df-sn 4632 df-pr 4634 |
This theorem is referenced by: pmtrsn 19552 topsn 22953 conncompid 23455 lfuhgr1v0e 29286 esumsnf 34045 cvmlift2lem9 35296 rrxtopn0b 46252 sge0sn 46335 |
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