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Mirrors > Home > ILE Home > Th. List > pwprss | GIF version |
Description: The power set of an unordered pair. (Contributed by Jim Kingdon, 13-Aug-2018.) |
Ref | Expression |
---|---|
pwprss | ⊢ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2617 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | 1 | elpr 3446 | . . . . 5 ⊢ (𝑥 ∈ {∅, {𝐴}} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐴})) |
3 | 1 | elpr 3446 | . . . . 5 ⊢ (𝑥 ∈ {{𝐵}, {𝐴, 𝐵}} ↔ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) |
4 | 2, 3 | orbi12i 714 | . . . 4 ⊢ ((𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}) ↔ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵}))) |
5 | ssprr 3577 | . . . 4 ⊢ (((𝑥 = ∅ ∨ 𝑥 = {𝐴}) ∨ (𝑥 = {𝐵} ∨ 𝑥 = {𝐴, 𝐵})) → 𝑥 ⊆ {𝐴, 𝐵}) | |
6 | 4, 5 | sylbi 119 | . . 3 ⊢ ((𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}}) → 𝑥 ⊆ {𝐴, 𝐵}) |
7 | elun 3127 | . . 3 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ↔ (𝑥 ∈ {∅, {𝐴}} ∨ 𝑥 ∈ {{𝐵}, {𝐴, 𝐵}})) | |
8 | 1 | elpw 3415 | . . 3 ⊢ (𝑥 ∈ 𝒫 {𝐴, 𝐵} ↔ 𝑥 ⊆ {𝐴, 𝐵}) |
9 | 6, 7, 8 | 3imtr4i 199 | . 2 ⊢ (𝑥 ∈ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) → 𝑥 ∈ 𝒫 {𝐴, 𝐵}) |
10 | 9 | ssriv 3016 | 1 ⊢ ({∅, {𝐴}} ∪ {{𝐵}, {𝐴, 𝐵}}) ⊆ 𝒫 {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∨ wo 662 = wceq 1287 ∈ wcel 1436 ∪ cun 2984 ⊆ wss 2986 ∅c0 3272 𝒫 cpw 3409 {csn 3425 {cpr 3426 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-v 2616 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 |
This theorem is referenced by: pwpwpw0ss 3628 ord3ex 3992 |
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