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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 4298 | . . 3 ⊢ (𝑥 ⊆ {𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) | |
2 | 1 | abbii 2726 | . 2 ⊢ {𝑥 ∣ 𝑥 ⊆ {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} |
3 | df-pw 4110 | . 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}} | |
4 | dfpr2 4143 | . 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} | |
5 | 2, 3, 4 | 3eqtr4i 2642 | 1 ⊢ 𝒫 {𝐴} = {∅, {𝐴}} |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 = wceq 1475 {cab 2596 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-pr 4128 |
This theorem is referenced by: pmtrsn 17762 topsn 20550 concompid 21044 usgra1v 25919 esumsnf 29453 cvmlift2lem9 30547 rrxtopn0b 39192 sge0sn 39272 lfuhgr1v0e 40480 |
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