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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 4758 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 𝒫 cpw 4541 {cpr 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-un 3943 df-in 3945 df-ss 3954 df-pw 4543 df-sn 4570 df-pr 4572 |
This theorem is referenced by: altxpsspw 33440 |
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