Metamath Proof Explorer

Theorem prnebg

Description: A (proper) pair is not equal to another (maybe improper) pair if and only if an element of the first pair is not contained in the second pair. (Contributed by Alexander van der Vekens, 16-Jan-2018)

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

Proof

Step Hyp Ref Expression
1 prneimg ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ) → ( ( ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∨ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) → { 𝐴 , 𝐵 } ≠ { 𝐶 , 𝐷 } ) )
2 1 3adant3 ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ∧ 𝐴 ≠ 𝐵 ) → ( ( ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∨ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) → { 𝐴 , 𝐵 } ≠ { 𝐶 , 𝐷 } ) )
3 ioran ⊢ ( ¬ ( ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∨ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) ↔ ( ¬ ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∧ ¬ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) )
4 ianor ⊢ ( ¬ ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ↔ ( ¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷 ) )
5 nne ⊢ ( ¬ 𝐴 ≠ 𝐶 ↔ 𝐴 = 𝐶 )
6 nne ⊢ ( ¬ 𝐴 ≠ 𝐷 ↔ 𝐴 = 𝐷 )
7 5 6 orbi12i ⊢ ( ( ¬ 𝐴 ≠ 𝐶 ∨ ¬ 𝐴 ≠ 𝐷 ) ↔ ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) )
8 4 7 bitri ⊢ ( ¬ ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ↔ ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) )
9 ianor ⊢ ( ¬ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ↔ ( ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷 ) )
10 nne ⊢ ( ¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶 )
11 nne ⊢ ( ¬ 𝐵 ≠ 𝐷 ↔ 𝐵 = 𝐷 )
12 10 11 orbi12i ⊢ ( ( ¬ 𝐵 ≠ 𝐶 ∨ ¬ 𝐵 ≠ 𝐷 ) ↔ ( 𝐵 = 𝐶 ∨ 𝐵 = 𝐷 ) )
13 9 12 bitri ⊢ ( ¬ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ↔ ( 𝐵 = 𝐶 ∨ 𝐵 = 𝐷 ) )
14 8 13 anbi12i ⊢ ( ( ¬ ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∧ ¬ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) ↔ ( ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) ∧ ( 𝐵 = 𝐶 ∨ 𝐵 = 𝐷 ) ) )
15 3 14 bitri ⊢ ( ¬ ( ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∨ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) ↔ ( ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) ∧ ( 𝐵 = 𝐶 ∨ 𝐵 = 𝐷 ) ) )
16 anddi ⊢ ( ( ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) ∧ ( 𝐵 = 𝐶 ∨ 𝐵 = 𝐷 ) ) ↔ ( ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ∨ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐷 ) ) ) )
17 eqtr3 ⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) → 𝐴 = 𝐵 )
18 eqneqall ⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≠ 𝐵 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
19 17 18 syl ⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) → ( 𝐴 ≠ 𝐵 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
20 preq12 ⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } )
21 20 a1d ⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝐴 ≠ 𝐵 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
22 19 21 jaoi ⊢ ( ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) → ( 𝐴 ≠ 𝐵 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
23 preq12 ⊢ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → { 𝐴 , 𝐵 } = { 𝐷 , 𝐶 } )
24 prcom ⊢ { 𝐷 , 𝐶 } = { 𝐶 , 𝐷 }
25 23 24 eqtrdi ⊢ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } )
26 25 a1d ⊢ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) → ( 𝐴 ≠ 𝐵 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
27 eqtr3 ⊢ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐷 ) → 𝐴 = 𝐵 )
28 27 18 syl ⊢ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐷 ) → ( 𝐴 ≠ 𝐵 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
29 26 28 jaoi ⊢ ( ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐷 ) ) → ( 𝐴 ≠ 𝐵 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
30 22 29 jaoi ⊢ ( ( ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ∨ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐷 ) ) ) → ( 𝐴 ≠ 𝐵 → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
31 30 com12 ⊢ ( 𝐴 ≠ 𝐵 → ( ( ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ∨ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐷 ) ) ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
32 31 3ad2ant3 ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ∧ 𝐴 ≠ 𝐵 ) → ( ( ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) ) ∨ ( ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐶 ) ∨ ( 𝐴 = 𝐷 ∧ 𝐵 = 𝐷 ) ) ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
33 16 32 syl5bi ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ∧ 𝐴 ≠ 𝐵 ) → ( ( ( 𝐴 = 𝐶 ∨ 𝐴 = 𝐷 ) ∧ ( 𝐵 = 𝐶 ∨ 𝐵 = 𝐷 ) ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
34 15 33 syl5bi ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ∧ 𝐴 ≠ 𝐵 ) → ( ¬ ( ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∨ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) → { 𝐴 , 𝐵 } = { 𝐶 , 𝐷 } ) )
35 34 necon1ad ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ∧ 𝐴 ≠ 𝐵 ) → ( { 𝐴 , 𝐵 } ≠ { 𝐶 , 𝐷 } → ( ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∨ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) ) )
36 2 35 impbid ⊢ ( ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ) ∧ 𝐴 ≠ 𝐵 ) → ( ( ( 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷 ) ∨ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ) ↔ { 𝐴 , 𝐵 } ≠ { 𝐶 , 𝐷 } ) )