Metamath Proof Explorer

Theorem cvrval

Description: Binary relation expressing B covers A , which means that B is larger than A and there is nothing in between. Definition 3.2.18 of PtakPulmannova p. 68. ( cvbr analog.) (Contributed by NM, 18-Sep-2011)

Ref Expression
Hypotheses cvrfval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cvrfval.s ⊢ < = ( lt ‘ 𝐾 )
cvrfval.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
Assertion cvrval ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 cvrfval.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cvrfval.s ⊢ < = ( lt ‘ 𝐾 )
3 cvrfval.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
4 1 2 3 cvrfval ⊢ ( 𝐾 ∈ 𝐴 → 𝐶 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) } )
5 3anass ⊢ ( ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ↔ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) )
6 5 opabbii ⊢ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) } = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) }
7 4 6 eqtrdi ⊢ ( 𝐾 ∈ 𝐴 → 𝐶 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) } )
8 7 breqd ⊢ ( 𝐾 ∈ 𝐴 → ( 𝑋 𝐶 𝑌 ↔ 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) } 𝑌 ) )
9 8 3ad2ant1 ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) } 𝑌 ) )
10 df-br ⊢ ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) } 𝑌 ↔ ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) } )
11 breq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 < 𝑦 ↔ 𝑋 < 𝑦 ) )
12 breq1 ⊢ ( 𝑥 = 𝑋 → ( 𝑥 < 𝑧 ↔ 𝑋 < 𝑧 ) )
13 12 anbi1d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ↔ ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑦 ) ) )
14 13 rexbidv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ↔ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑦 ) ) )
15 14 notbid ⊢ ( 𝑥 = 𝑋 → ( ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ↔ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑦 ) ) )
16 11 15 anbi12d ⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ↔ ( 𝑋 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) )
17 breq2 ⊢ ( 𝑦 = 𝑌 → ( 𝑋 < 𝑦 ↔ 𝑋 < 𝑌 ) )
18 breq2 ⊢ ( 𝑦 = 𝑌 → ( 𝑧 < 𝑦 ↔ 𝑧 < 𝑌 ) )
19 18 anbi2d ⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑦 ) ↔ ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) )
20 19 rexbidv ⊢ ( 𝑦 = 𝑌 → ( ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑦 ) ↔ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) )
21 20 notbid ⊢ ( 𝑦 = 𝑌 → ( ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑦 ) ↔ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) )
22 17 21 anbi12d ⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )
23 16 22 opelopab2 ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) } ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )
24 10 23 syl5bb ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) } 𝑌 ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )
25 24 3adant1 ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ∧ ( 𝑥 < 𝑦 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) ) ) } 𝑌 ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )
26 9 25 bitrd ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )