Theorem sotr3 5632

Description: Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)

Assertion

Ref	Expression
sotr3	⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))

Proof of Theorem sotr3

Step	Hyp	Ref	Expression
1		simp3 1138	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → 𝑍 ∈ 𝐴)
2		simp2 1137	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → 𝑌 ∈ 𝐴)
3	1, 2	jca 511	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → (𝑍 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴))
4		sotric 5621	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑍 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌 ∨ 𝑌𝑅𝑍)))
5	3, 4	sylan2 593	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌 ∨ 𝑌𝑅𝑍)))
6	5	con2bid 354	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
7	6	adantr 480	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
8		breq2 5146	. . . . . 6 ⊢ (𝑍 = 𝑌 → (𝑋𝑅𝑍 ↔ 𝑋𝑅𝑌))
9	8	biimprcd 250	. . . . 5 ⊢ (𝑋𝑅𝑌 → (𝑍 = 𝑌 → 𝑋𝑅𝑍))
10	9	adantl 481	. . . 4 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (𝑍 = 𝑌 → 𝑋𝑅𝑍))
11		sotr 5616	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
12	11	expdimp 452	. . . 4 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (𝑌𝑅𝑍 → 𝑋𝑅𝑍))
13	10, 12	jaod 859	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
14	7, 13	sylbird 260	. 2 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (¬ 𝑍𝑅𝑌 → 𝑋𝑅𝑍))
15	14	expimpd 453	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   class class class wbr 5142   Or wor 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-po 5591  df-so 5592

This theorem is referenced by:  nosupbnd2  27762  noinfbnd1  27775  sltletr  27802