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Metamath Proof Explorer

Theorem prproe

Description: For an element of a proper unordered pair of elements of a class V , there is another (different) element of the class V which is an element of the proper pair. (Contributed by AV, 18-Dec-2021)

Ref Expression
Assertion prproe ( ( 𝐶 ∈ { 𝐴 , 𝐵 } ∧ 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 𝐶 } ) 𝑣 ∈ { 𝐴 , 𝐵 } )

Proof

Step Hyp Ref Expression
1 elpri ⊢ ( 𝐶 ∈ { 𝐴 , 𝐵 } → ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) )
2 eleq1 ⊢ ( 𝑣 = 𝐵 → ( 𝑣 ∈ { 𝐴 , 𝐵 } ↔ 𝐵 ∈ { 𝐴 , 𝐵 } ) )
3 simprrr ⊢ ( ( 𝐶 = 𝐴 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → 𝐵 ∈ 𝑉 )
4 necom ⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴 )
5 neeq2 ⊢ ( 𝐴 = 𝐶 → ( 𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶 ) )
6 5 eqcoms ⊢ ( 𝐶 = 𝐴 → ( 𝐵 ≠ 𝐴 ↔ 𝐵 ≠ 𝐶 ) )
7 6 biimpcd ⊢ ( 𝐵 ≠ 𝐴 → ( 𝐶 = 𝐴 → 𝐵 ≠ 𝐶 ) )
8 4 7 sylbi ⊢ ( 𝐴 ≠ 𝐵 → ( 𝐶 = 𝐴 → 𝐵 ≠ 𝐶 ) )
9 8 adantr ⊢ ( ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ( 𝐶 = 𝐴 → 𝐵 ≠ 𝐶 ) )
10 9 impcom ⊢ ( ( 𝐶 = 𝐴 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → 𝐵 ≠ 𝐶 )
11 3 10 eldifsnd ⊢ ( ( 𝐶 = 𝐴 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → 𝐵 ∈ ( 𝑉 ∖ { 𝐶 } ) )
12 prid2g ⊢ ( 𝐵 ∈ 𝑉 → 𝐵 ∈ { 𝐴 , 𝐵 } )
13 12 adantl ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐵 ∈ { 𝐴 , 𝐵 } )
14 13 ad2antll ⊢ ( ( 𝐶 = 𝐴 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → 𝐵 ∈ { 𝐴 , 𝐵 } )
15 2 11 14 rspcedvdw ⊢ ( ( 𝐶 = 𝐴 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 𝐶 } ) 𝑣 ∈ { 𝐴 , 𝐵 } )
16 15 ex ⊢ ( 𝐶 = 𝐴 → ( ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 𝐶 } ) 𝑣 ∈ { 𝐴 , 𝐵 } ) )
17 eleq1 ⊢ ( 𝑣 = 𝐴 → ( 𝑣 ∈ { 𝐴 , 𝐵 } ↔ 𝐴 ∈ { 𝐴 , 𝐵 } ) )
18 simprrl ⊢ ( ( 𝐶 = 𝐵 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → 𝐴 ∈ 𝑉 )
19 neeq2 ⊢ ( 𝐵 = 𝐶 → ( 𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶 ) )
20 19 eqcoms ⊢ ( 𝐶 = 𝐵 → ( 𝐴 ≠ 𝐵 ↔ 𝐴 ≠ 𝐶 ) )
21 20 biimpcd ⊢ ( 𝐴 ≠ 𝐵 → ( 𝐶 = 𝐵 → 𝐴 ≠ 𝐶 ) )
22 21 adantr ⊢ ( ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ( 𝐶 = 𝐵 → 𝐴 ≠ 𝐶 ) )
23 22 impcom ⊢ ( ( 𝐶 = 𝐵 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → 𝐴 ≠ 𝐶 )
24 18 23 eldifsnd ⊢ ( ( 𝐶 = 𝐵 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → 𝐴 ∈ ( 𝑉 ∖ { 𝐶 } ) )
25 prid1g ⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ { 𝐴 , 𝐵 } )
26 25 adantr ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ { 𝐴 , 𝐵 } )
27 26 ad2antll ⊢ ( ( 𝐶 = 𝐵 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → 𝐴 ∈ { 𝐴 , 𝐵 } )
28 17 24 27 rspcedvdw ⊢ ( ( 𝐶 = 𝐵 ∧ ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 𝐶 } ) 𝑣 ∈ { 𝐴 , 𝐵 } )
29 28 ex ⊢ ( 𝐶 = 𝐵 → ( ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 𝐶 } ) 𝑣 ∈ { 𝐴 , 𝐵 } ) )
30 16 29 jaoi ⊢ ( ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) → ( ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 𝐶 } ) 𝑣 ∈ { 𝐴 , 𝐵 } ) )
31 1 30 syl ⊢ ( 𝐶 ∈ { 𝐴 , 𝐵 } → ( ( 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 𝐶 } ) 𝑣 ∈ { 𝐴 , 𝐵 } ) )
32 31 3impib ⊢ ( ( 𝐶 ∈ { 𝐴 , 𝐵 } ∧ 𝐴 ≠ 𝐵 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) → ∃ 𝑣 ∈ ( 𝑉 ∖ { 𝐶 } ) 𝑣 ∈ { 𝐴 , 𝐵 } )