Description: An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004) (Revised by Mario Carneiro, 26-Apr-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | opnz | ⊢ ( 〈 𝐴 , 𝐵 〉 ≠ ∅ ↔ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprc | ⊢ ( ¬ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 〈 𝐴 , 𝐵 〉 = ∅ ) | |
2 | 1 | necon1ai | ⊢ ( 〈 𝐴 , 𝐵 〉 ≠ ∅ → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) |
3 | dfopg | ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 〈 𝐴 , 𝐵 〉 = { { 𝐴 } , { 𝐴 , 𝐵 } } ) | |
4 | snex | ⊢ { 𝐴 } ∈ V | |
5 | 4 | prnz | ⊢ { { 𝐴 } , { 𝐴 , 𝐵 } } ≠ ∅ |
6 | 5 | a1i | ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → { { 𝐴 } , { 𝐴 , 𝐵 } } ≠ ∅ ) |
7 | 3 6 | eqnetrd | ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → 〈 𝐴 , 𝐵 〉 ≠ ∅ ) |
8 | 2 7 | impbii | ⊢ ( 〈 𝐴 , 𝐵 〉 ≠ ∅ ↔ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) |