Theorem sotr2 5535

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

Assertion

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

Proof of Theorem sotr2

Step	Hyp	Ref	Expression
1		sotric 5531	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
2	1	ancom2s 647	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
3	2	3adantr3 1170	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
4	3	con2bid 355	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
5		breq1 5077	. . . . . 6 ⊢ (𝐶 = 𝐵 → (𝐶𝑅𝐷 ↔ 𝐵𝑅𝐷))
6	5	biimpd 228	. . . . 5 ⊢ (𝐶 = 𝐵 → (𝐶𝑅𝐷 → 𝐵𝑅𝐷))
7	6	a1i 11	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 = 𝐵 → (𝐶𝑅𝐷 → 𝐵𝑅𝐷)))
8		sotr 5527	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))
9	8	expd 416	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵𝑅𝐶 → (𝐶𝑅𝐷 → 𝐵𝑅𝐷)))
10	7, 9	jaod 856	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) → (𝐶𝑅𝐷 → 𝐵𝑅𝐷)))
11	4, 10	sylbird 259	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ 𝐶𝑅𝐵 → (𝐶𝑅𝐷 → 𝐵𝑅𝐷)))
12	11	impd 411	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((¬ 𝐶𝑅𝐵 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   class class class wbr 5074   Or wor 5502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-po 5503  df-so 5504

This theorem is referenced by:  erdszelem8  33160  nosupbnd1  33917  noinfbnd2  33934  slelttr  33960