| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 4823 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶))) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ ({𝐴, 𝐵} ⊆ 𝒫 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-pw 4602 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: altxpsspw 35978 |
| Copyright terms: Public domain | W3C validator |