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Mirrors > Home > MPE Home > Th. List > pwpr | Structured version Visualization version GIF version |
Description: The power set of an unordered pair. (Contributed by NM, 1-May-2009.) |
Ref | Expression |
---|---|
pwpr | ⊢ 𝒫 {𝐴, 𝐵} = ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspr 4769 | . . . 4 ⊢ (𝑥 ⊆ {𝐴, 𝐵} ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))) | |
2 | vex 3500 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | elpr 4593 | . . . . 5 ⊢ (𝑥 ∈ {∅, {𝐴}} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) |
4 | 2 | elpr 4593 | . . . . 5 ⊢ (𝑥 ∈ {{𝐵}, {𝐴, 𝐵}} ↔ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) |
5 | 3, 4 | orbi12i 911 | . . . 4 ⊢ ((𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))) |
6 | 1, 5 | bitr4i 280 | . . 3 ⊢ (𝑥 ⊆ {𝐴, 𝐵} ↔ (𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}})) |
7 | velpw 4547 | . . 3 ⊢ (𝑥 ∈ 𝒫 {𝐴, 𝐵} ↔ 𝑥 ⊆ {𝐴, 𝐵}) | |
8 | elun 4128 | . . 3 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ (𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}})) | |
9 | 6, 7, 8 | 3bitr4i 305 | . 2 ⊢ (𝑥 ∈ 𝒫 {𝐴, 𝐵} ↔ 𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}})) |
10 | 9 | eqriv 2821 | 1 ⊢ 𝒫 {𝐴, 𝐵} = ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 843 = wceq 1536 ∈ wcel 2113 ∪ cun 3937 ⊆ wss 3939 ∅c0 4294 𝒫 cpw 4542 {csn 4570 {cpr 4572 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-pw 4544 df-sn 4571 df-pr 4573 |
This theorem is referenced by: pwpwpw0 4837 ord3ex 5291 hash2pwpr 13837 pr2pwpr 13840 prsiga 31394 prsal 42610 |
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