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Mirrors > Home > MPE Home > Th. List > prsspw | Structured version Visualization version 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.) (Proof shortened by OpenAI, 25-Mar-2020.) |
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 | . 2 ⊢ 𝐴 ∈ V | |
2 | prsspw.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | prsspwg 4667 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))) | |
4 | 1, 2, 3 | mp2an 688 | 1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∈ wcel 2081 Vcvv 3437 ⊆ wss 3863 𝒫 cpw 4457 {cpr 4478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-un 3868 df-in 3870 df-ss 3878 df-pw 4459 df-sn 4477 df-pr 4479 |
This theorem is referenced by: altxpsspw 33054 |
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