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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 4719 | . . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) | |
2 | 1 | abbii 2863 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} |
3 | df-pw 4499 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
4 | dfpr2 4544 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
5 | 2, 3, 4 | 3eqtr4i 2831 | 1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 844 = wceq 1538 {cab 2776 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 {cpr 4527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-pw 4499 df-sn 4526 df-pr 4528 |
This theorem is referenced by: pmtrsn 18639 topsn 21536 conncompid 22036 lfuhgr1v0e 27044 esumsnf 31433 cvmlift2lem9 32671 rrxtopn0b 42938 sge0sn 43018 |
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