Metamath Proof Explorer


Theorem po2ne

Description: Two sets related by a partial order are not equal. (Contributed by AV, 13-Mar-2023)

Ref Expression
Assertion po2ne ( ( 𝑅 Po 𝑉 ∧ ( 𝐴𝑉𝐵𝑉 ) ∧ 𝐴 𝑅 𝐵 ) → 𝐴𝐵 )

Proof

Step Hyp Ref Expression
1 breq1 ( 𝐴 = 𝐵 → ( 𝐴 𝑅 𝐵𝐵 𝑅 𝐵 ) )
2 poirr ( ( 𝑅 Po 𝑉𝐵𝑉 ) → ¬ 𝐵 𝑅 𝐵 )
3 2 adantrl ( ( 𝑅 Po 𝑉 ∧ ( 𝐴𝑉𝐵𝑉 ) ) → ¬ 𝐵 𝑅 𝐵 )
4 3 pm2.21d ( ( 𝑅 Po 𝑉 ∧ ( 𝐴𝑉𝐵𝑉 ) ) → ( 𝐵 𝑅 𝐵𝐴𝐵 ) )
5 4 ex ( 𝑅 Po 𝑉 → ( ( 𝐴𝑉𝐵𝑉 ) → ( 𝐵 𝑅 𝐵𝐴𝐵 ) ) )
6 5 com13 ( 𝐵 𝑅 𝐵 → ( ( 𝐴𝑉𝐵𝑉 ) → ( 𝑅 Po 𝑉𝐴𝐵 ) ) )
7 1 6 syl6bi ( 𝐴 = 𝐵 → ( 𝐴 𝑅 𝐵 → ( ( 𝐴𝑉𝐵𝑉 ) → ( 𝑅 Po 𝑉𝐴𝐵 ) ) ) )
8 7 com24 ( 𝐴 = 𝐵 → ( 𝑅 Po 𝑉 → ( ( 𝐴𝑉𝐵𝑉 ) → ( 𝐴 𝑅 𝐵𝐴𝐵 ) ) ) )
9 8 3impd ( 𝐴 = 𝐵 → ( ( 𝑅 Po 𝑉 ∧ ( 𝐴𝑉𝐵𝑉 ) ∧ 𝐴 𝑅 𝐵 ) → 𝐴𝐵 ) )
10 ax-1 ( 𝐴𝐵 → ( ( 𝑅 Po 𝑉 ∧ ( 𝐴𝑉𝐵𝑉 ) ∧ 𝐴 𝑅 𝐵 ) → 𝐴𝐵 ) )
11 9 10 pm2.61ine ( ( 𝑅 Po 𝑉 ∧ ( 𝐴𝑉𝐵𝑉 ) ∧ 𝐴 𝑅 𝐵 ) → 𝐴𝐵 )