Theorem sotr2 5574

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 5570	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
2	1	ancom2s 651	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
3	2	3adantr3 1173	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
4	3	con2bid 354	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
5		breq1 5103	. . . . . 6 ⊢ (𝐶 = 𝐵 → (𝐶𝑅𝐷 ↔ 𝐵𝑅𝐷))
6	5	biimpd 229	. . . . 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 5565	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))
9	8	expd 415	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐵𝑅𝐶 → (𝐶𝑅𝐷 → 𝐵𝑅𝐷)))
10	7, 9	jaod 860	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) → (𝐶𝑅𝐷 → 𝐵𝑅𝐷)))
11	4, 10	sylbird 260	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (¬ 𝐶𝑅𝐵 → (𝐶𝑅𝐷 → 𝐵𝑅𝐷)))
12	11	impd 410	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((¬ 𝐶𝑅𝐵 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5100   Or wor 5539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-po 5540  df-so 5541

This theorem is referenced by:  nosupbnd1  27694  noinfbnd2  27711  leltstr  27741  erdszelem8  35414