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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 4543 | . . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) | |
2 | 1 | abbii 2914 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} |
3 | df-pw 4349 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
4 | dfpr2 4385 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
5 | 2, 3, 4 | 3eqtr4i 2829 | 1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 874 = wceq 1653 {cab 2783 ⊆ wss 3767 ∅c0 4113 𝒫 cpw 4347 {csn 4366 {cpr 4368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-pw 4349 df-sn 4367 df-pr 4369 |
This theorem is referenced by: pmtrsn 18249 topsn 21061 conncompid 21560 lfuhgr1v0e 26480 esumsnf 30634 cvmlift2lem9 31802 rrxtopn0b 41247 sge0sn 41327 |
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