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Metamath Proof Explorer

Theorem cvrnbtwn2

Description: The covers relation implies no in-betweenness. ( cvnbtwn2 analog.) (Contributed by NM, 17-Nov-2011)

Ref Expression
Hypotheses cvrletr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cvrletr.l ⊢ ≤ = ( le ‘ 𝐾 )
cvrletr.s ⊢ < = ( lt ‘ 𝐾 )
cvrletr.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
Assertion cvrnbtwn2 ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) ↔ 𝑍 = 𝑌 ) )

Proof

Step Hyp Ref Expression
1 cvrletr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cvrletr.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cvrletr.s ⊢ < = ( lt ‘ 𝐾 )
4 cvrletr.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
5 1 3 4 cvrnbtwn ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) )
6 5 3expia ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑌 → ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
7 iman ⊢ ( ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) → 𝑍 = 𝑌 ) ↔ ¬ ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) ∧ ¬ 𝑍 = 𝑌 ) )
8 anass ⊢ ( ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) ∧ ¬ 𝑍 = 𝑌 ) ↔ ( 𝑋 < 𝑍 ∧ ( 𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌 ) ) )
9 simpl ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Poset )
10 simpr3 ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
11 simpr2 ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
12 2 3 pltval ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑍 < 𝑌 ↔ ( 𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌 ) ) )
13 9 10 11 12 syl3anc ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑍 < 𝑌 ↔ ( 𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌 ) ) )
14 df-ne ⊢ ( 𝑍 ≠ 𝑌 ↔ ¬ 𝑍 = 𝑌 )
15 14 anbi2i ⊢ ( ( 𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌 ) ↔ ( 𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌 ) )
16 13 15 bitrdi ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑍 < 𝑌 ↔ ( 𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌 ) ) )
17 16 anbi2d ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ↔ ( 𝑋 < 𝑍 ∧ ( 𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌 ) ) ) )
18 8 17 bitr4id ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) ∧ ¬ 𝑍 = 𝑌 ) ↔ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
19 18 notbid ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ¬ ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) ∧ ¬ 𝑍 = 𝑌 ) ↔ ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ) )
20 7 19 bitr2id ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ¬ ( 𝑋 < 𝑍 ∧ 𝑍 < 𝑌 ) ↔ ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) → 𝑍 = 𝑌 ) ) )
21 6 20 sylibd ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑌 → ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) → 𝑍 = 𝑌 ) ) )
22 21 3impia ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) → 𝑍 = 𝑌 ) )
23 1 3 4 cvrlt ⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 < 𝑌 )
24 23 ex ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 → 𝑋 < 𝑌 ) )
25 24 3adant3r3 ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑌 → 𝑋 < 𝑌 ) )
26 25 3impia ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 < 𝑌 )
27 breq2 ⊢ ( 𝑍 = 𝑌 → ( 𝑋 < 𝑍 ↔ 𝑋 < 𝑌 ) )
28 26 27 syl5ibrcom ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ( 𝑍 = 𝑌 → 𝑋 < 𝑍 ) )
29 1 2 posref ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ≤ 𝑌 )
30 29 3ad2antr2 ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ≤ 𝑌 )
31 breq1 ⊢ ( 𝑍 = 𝑌 → ( 𝑍 ≤ 𝑌 ↔ 𝑌 ≤ 𝑌 ) )
32 30 31 syl5ibrcom ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑍 = 𝑌 → 𝑍 ≤ 𝑌 ) )
33 32 3adant3 ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ( 𝑍 = 𝑌 → 𝑍 ≤ 𝑌 ) )
34 28 33 jcad ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ( 𝑍 = 𝑌 → ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) ) )
35 22 34 impbid ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → ( ( 𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌 ) ↔ 𝑍 = 𝑌 ) )