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Mirrors > Home > NFE Home > Th. List > pwpr | GIF version |
Description: The power set of an unordered pair. (Contributed by NM, 1-May-2009.) |
Ref | Expression |
---|---|
pwpr | ⊢ ℘{A, B} = ({∅, {A}} ∪ {{B}, {A, B}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspr 3869 | . . . 4 ⊢ (x ⊆ {A, B} ↔ ((x = ∅ ∨ x = {A}) ∨ (x = {B} ∨ x = {A, B}))) | |
2 | vex 2862 | . . . . . 6 ⊢ x ∈ V | |
3 | 2 | elpr 3751 | . . . . 5 ⊢ (x ∈ {∅, {A}} ↔ (x = ∅ ∨ x = {A})) |
4 | 2 | elpr 3751 | . . . . 5 ⊢ (x ∈ {{B}, {A, B}} ↔ (x = {B} ∨ x = {A, B})) |
5 | 3, 4 | orbi12i 507 | . . . 4 ⊢ ((x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}}) ↔ ((x = ∅ ∨ x = {A}) ∨ (x = {B} ∨ x = {A, B}))) |
6 | 1, 5 | bitr4i 243 | . . 3 ⊢ (x ⊆ {A, B} ↔ (x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}})) |
7 | 2 | elpw 3728 | . . 3 ⊢ (x ∈ ℘{A, B} ↔ x ⊆ {A, B}) |
8 | elun 3220 | . . 3 ⊢ (x ∈ ({∅, {A}} ∪ {{B}, {A, B}}) ↔ (x ∈ {∅, {A}} ∨ x ∈ {{B}, {A, B}})) | |
9 | 6, 7, 8 | 3bitr4i 268 | . 2 ⊢ (x ∈ ℘{A, B} ↔ x ∈ ({∅, {A}} ∪ {{B}, {A, B}})) |
10 | 9 | eqriv 2350 | 1 ⊢ ℘{A, B} = ({∅, {A}} ∪ {{B}, {A, B}}) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 357 = wceq 1642 ∈ wcel 1710 ∪ cun 3207 ⊆ wss 3257 ∅c0 3550 ℘cpw 3722 {csn 3737 {cpr 3738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-pr 3742 |
This theorem is referenced by: pwpwpw0 3885 |
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