Theorem sotr3 5611

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 1154	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → 𝑍 ∈ 𝐴)
2		simp2 1153	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → 𝑌 ∈ 𝐴)
3	1, 2	jca 520	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → (𝑍 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴))
4		sotric 5600	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑍 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌 ∨ 𝑌𝑅𝑍)))
5	3, 4	sylan2 604	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌 ∨ 𝑌𝑅𝑍)))
6	5	con2bid 357	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
7	6	adantr 485	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
8		breq2 5117	. . . . . 6 ⊢ (𝑍 = 𝑌 → (𝑋𝑅𝑍 ↔ 𝑋𝑅𝑌))
9	8	biimprcd 253	. . . . 5 ⊢ (𝑋𝑅𝑌 → (𝑍 = 𝑌 → 𝑋𝑅𝑍))
10	9	adantl 486	. . . 4 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (𝑍 = 𝑌 → 𝑋𝑅𝑍))
11		sotr 5595	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
12	11	expdimp 457	. . . 4 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (𝑌𝑅𝑍 → 𝑋𝑅𝑍))
13	10, 12	jaod 872	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
14	7, 13	sylbird 263	. 2 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (¬ 𝑍𝑅𝑌 → 𝑋𝑅𝑍))
15	14	expimpd 458	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   class class class wbr 5113   Or wor 5569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-po 5570  df-so 5571

This theorem is referenced by:  nosupbnd2  27845  noinfbnd1  27858  ltlestr  27889