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Mirrors > Home > ILE Home > Th. List > prsspw | GIF version |
Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
prsspw.1 | ⊢ 𝐴 ∈ V |
prsspw.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
prsspw | ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prsspw.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | prsspw.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | prss 3684 | . 2 ⊢ ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ {𝐴, 𝐵} ⊆ 𝒫 𝐶) |
4 | 1 | elpw 3521 | . . 3 ⊢ (𝐴 ∈ 𝒫 𝐶 ↔ 𝐴 ⊆ 𝐶) |
5 | 2 | elpw 3521 | . . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 ↔ 𝐵 ⊆ 𝐶) |
6 | 4, 5 | anbi12i 456 | . 2 ⊢ ((𝐴 ∈ 𝒫 𝐶 ∧ 𝐵 ∈ 𝒫 𝐶) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
7 | 3, 6 | bitr3i 185 | 1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 𝒫 cpw 3515 {cpr 3533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 |
This theorem is referenced by: (None) |
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