Theorem tsrlin 18218

Description: A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)

Hypothesis

Ref	Expression
istsr.1	⊢ 𝑋 = dom 𝑅

Assertion

Ref	Expression
tsrlin	⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))

Proof of Theorem tsrlin

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		istsr.1	. . . . 5 ⊢ 𝑋 = dom 𝑅
2	1	istsr2 18217	. . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
3	2	simprbi 496	. . 3 ⊢ (𝑅 ∈ TosetRel → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
4		breq1 5073	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
5		breq2 5074	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
6	4, 5	orbi12d 915	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴)))
7		breq2 5074	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
8		breq1 5073	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
9	7, 8	orbi12d 915	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴)))
10	6, 9	rspc2v 3562	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴)))
11	3, 10	syl5com 31	. 2 ⊢ (𝑅 ∈ TosetRel → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴)))
12	11	3impib 1114	1 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070  dom cdm 5580  PosetRelcps 18197   TosetRel ctsr 18198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-tsr 18200

This theorem is referenced by:  tsrlemax  18219  ordtrest2lem  22262  ordthauslem  22442  ordthaus  22443