Theorem sotr3 31656

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 1063	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → 𝑍 ∈ 𝐴)
2		simp2 1062	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → 𝑌 ∈ 𝐴)
3	1, 2	jca 554	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴) → (𝑍 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴))
4		sotric 5061	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝑍 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌 ∨ 𝑌𝑅𝑍)))
5	3, 4	sylan2 491	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → (𝑍𝑅𝑌 ↔ ¬ (𝑍 = 𝑌 ∨ 𝑌𝑅𝑍)))
6	5	con2bid 344	. . . 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 4657	. . . . . 6 ⊢ (𝑍 = 𝑌 → (𝑋𝑅𝑍 ↔ 𝑋𝑅𝑌))
9	8	biimprcd 240	. . . . 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 5057	. . . . 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 395	. . 3 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → ((𝑍 = 𝑌 ∨ 𝑌𝑅𝑍) → 𝑋𝑅𝑍))
14	7, 13	sylbird 250	. 2 ⊢ (((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) ∧ 𝑋𝑅𝑌) → (¬ 𝑍𝑅𝑌 → 𝑋𝑅𝑍))
15	14	expimpd 629	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   class class class wbr 4653   Or wor 5034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-po 5035  df-so 5036

This theorem is referenced by:  nosupbnd2  31862  sltletr  31881