Metamath Proof Explorer


Theorem posasymb

Description: A poset ordering is asymmetric. (Contributed by NM, 21-Oct-2011)

Ref Expression
Hypotheses posi.b 𝐵 = ( Base ‘ 𝐾 )
posi.l = ( le ‘ 𝐾 )
Assertion posasymb ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 𝑌𝑌 𝑋 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 posi.b 𝐵 = ( Base ‘ 𝐾 )
2 posi.l = ( le ‘ 𝐾 )
3 simp1 ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵 ) → 𝐾 ∈ Poset )
4 simp2 ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵 ) → 𝑋𝐵 )
5 simp3 ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵 ) → 𝑌𝐵 )
6 1 2 posi ( ( 𝐾 ∈ Poset ∧ ( 𝑋𝐵𝑌𝐵𝑌𝐵 ) ) → ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑌𝑌 𝑋 ) → 𝑋 = 𝑌 ) ∧ ( ( 𝑋 𝑌𝑌 𝑌 ) → 𝑋 𝑌 ) ) )
7 3 4 5 5 6 syl13anc ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑋 ∧ ( ( 𝑋 𝑌𝑌 𝑋 ) → 𝑋 = 𝑌 ) ∧ ( ( 𝑋 𝑌𝑌 𝑌 ) → 𝑋 𝑌 ) ) )
8 7 simp2d ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 𝑌𝑌 𝑋 ) → 𝑋 = 𝑌 ) )
9 1 2 posref ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵 ) → 𝑋 𝑋 )
10 breq2 ( 𝑋 = 𝑌 → ( 𝑋 𝑋𝑋 𝑌 ) )
11 9 10 syl5ibcom ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵 ) → ( 𝑋 = 𝑌𝑋 𝑌 ) )
12 breq1 ( 𝑋 = 𝑌 → ( 𝑋 𝑋𝑌 𝑋 ) )
13 9 12 syl5ibcom ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵 ) → ( 𝑋 = 𝑌𝑌 𝑋 ) )
14 11 13 jcad ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵 ) → ( 𝑋 = 𝑌 → ( 𝑋 𝑌𝑌 𝑋 ) ) )
15 14 3adant3 ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 = 𝑌 → ( 𝑋 𝑌𝑌 𝑋 ) ) )
16 8 15 impbid ( ( 𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 𝑌𝑌 𝑋 ) ↔ 𝑋 = 𝑌 ) )