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Metamath Proof Explorer

Theorem prneimg2

Description: Two pairs are not equal if their counterparts are not equal. (Contributed by AV, 5-Sep-2025)

Ref Expression
Assertion prneimg2 ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ) → ( { 𝐴 , 𝐵 } ≠ { 𝐶 , 𝐷 } ↔ ( ( 𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷 ) ∧ ( 𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶 ) ) ) )

Proof

Step Hyp Ref Expression
1 preq12bg ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ) → ( { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ↔ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) ) )
2 1 necon3abid ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ) → ( { 𝐴 , 𝐵 } ≠ { 𝐶 , 𝐷 } ↔ ¬ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) ) )
3 ioran ⊢ ( ¬ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) ↔ ( ¬ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∧ ¬ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) )
4 ianor ⊢ ( ¬ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ↔ ( ¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷 ) )
5 df-ne ⊢ ( 𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶 )
6 df-ne ⊢ ( 𝐵 ≠ 𝐷 ↔ ¬ 𝐵 = 𝐷 )
7 5 6 orbi12i ⊢ ( ( 𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷 ) ↔ ( ¬ 𝐴 = 𝐶 ∨ ¬ 𝐵 = 𝐷 ) )
8 4 7 bitr4i ⊢ ( ¬ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ↔ ( 𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷 ) )
9 ianor ⊢ ( ¬ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ↔ ( ¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐶 ) )
10 df-ne ⊢ ( 𝐴 ≠ 𝐷 ↔ ¬ 𝐴 = 𝐷 )
11 df-ne ⊢ ( 𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶 )
12 10 11 orbi12i ⊢ ( ( 𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶 ) ↔ ( ¬ 𝐴 = 𝐷 ∨ ¬ 𝐵 = 𝐶 ) )
13 9 12 bitr4i ⊢ ( ¬ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ↔ ( 𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶 ) )
14 8 13 anbi12i ⊢ ( ( ¬ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∧ ¬ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) ↔ ( ( 𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷 ) ∧ ( 𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶 ) ) )
15 3 14 bitri ⊢ ( ¬ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ) ↔ ( ( 𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷 ) ∧ ( 𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶 ) ) )
16 2 15 bitrdi ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ) → ( { 𝐴 , 𝐵 } ≠ { 𝐶 , 𝐷 } ↔ ( ( 𝐴 ≠ 𝐶 ∨ 𝐵 ≠ 𝐷 ) ∧ ( 𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐶 ) ) ) )