Theorem sotr3 5567

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 1144	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → 𝑍 ∈ 𝐴)
2		simp2 1143	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → 𝑌 ∈ 𝐴)
3	1, 2	jca 516	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → (𝑍 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴))
4		sotric 5556	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑍 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌 ∨ 𝑌𝑅𝑍)))
5	3, 4	sylan2 599	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌 ∨ 𝑌𝑅𝑍)))
6	5	con2bid 355	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
7	6	adantr 481	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) ↔ ¬ 𝑍𝑅𝑌))
8		breq2 5076	. . . . . 6 ⊢ (𝑍 = 𝑌 → (𝑋𝑅𝑍 ↔ 𝑋𝑅𝑌))
9	8	biimprcd 251	. . . . 5 ⊢ (𝑋𝑅𝑌 → (𝑍 = 𝑌 → 𝑋𝑅𝑍))
10	9	adantl 482	. . . 4 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (𝑍 = 𝑌 → 𝑋𝑅𝑍))
11		sotr 5551	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
12	11	expdimp 453	. . . 4 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (𝑌𝑅𝑍 → 𝑋𝑅𝑍))
13	10, 12	jaod 865	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
14	7, 13	sylbird 261	. 2 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (¬ 𝑍𝑅𝑌 → 𝑋𝑅𝑍))
15	14	expimpd 454	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   class class class wbr 5072   Or wor 5525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-po 5526  df-so 5527

This theorem is referenced by:  nosupbnd2  27698  noinfbnd1  27711  ltlestr  27742