Theorem sotr2

Description: A transitivity relation. (Read B <_ C and C < D implies B < D .) (Contributed by Mario Carneiro, 10-May-2013)

	Ref	Expression
Assertion	sotr2	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( ¬ 𝐶 𝑅 𝐵 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		sotric	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ) ) → ( 𝐶 𝑅 𝐵 ↔ ¬ ( 𝐶 = 𝐵 ∨ 𝐵 𝑅 𝐶 ) ) )
2	1	ancom2s	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐶 𝑅 𝐵 ↔ ¬ ( 𝐶 = 𝐵 ∨ 𝐵 𝑅 𝐶 ) ) )
3	2	3adantr3	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( 𝐶 𝑅 𝐵 ↔ ¬ ( 𝐶 = 𝐵 ∨ 𝐵 𝑅 𝐶 ) ) )
4	3	con2bid	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( 𝐶 = 𝐵 ∨ 𝐵 𝑅 𝐶 ) ↔ ¬ 𝐶 𝑅 𝐵 ) )
5		breq1	⊢ ( 𝐶 = 𝐵 → ( 𝐶 𝑅 𝐷 ↔ 𝐵 𝑅 𝐷 ) )
6	5	biimpd	⊢ ( 𝐶 = 𝐵 → ( 𝐶 𝑅 𝐷 → 𝐵 𝑅 𝐷 ) )
7	6	a1i	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( 𝐶 = 𝐵 → ( 𝐶 𝑅 𝐷 → 𝐵 𝑅 𝐷 ) ) )
8		sotr	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) )
9	8	expd	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( 𝐵 𝑅 𝐶 → ( 𝐶 𝑅 𝐷 → 𝐵 𝑅 𝐷 ) ) )
10	7 9	jaod	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( 𝐶 = 𝐵 ∨ 𝐵 𝑅 𝐶 ) → ( 𝐶 𝑅 𝐷 → 𝐵 𝑅 𝐷 ) ) )
11	4 10	sylbird	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ¬ 𝐶 𝑅 𝐵 → ( 𝐶 𝑅 𝐷 → 𝐵 𝑅 𝐷 ) ) )
12	11	impd	⊢ ( ( 𝑅 Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) ) → ( ( ¬ 𝐶 𝑅 𝐵 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) )