Theorem tsrlin 18483

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 18482	. . . 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 5092	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
5		breq2 5093	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
6	4, 5	orbi12d 918	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴)))
7		breq2 5093	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
8		breq1 5092	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
9	7, 8	orbi12d 918	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴)))
10	6, 9	rspc2v 3586	. . 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 1116	1 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2110  ∀wral 3045   class class class wbr 5089  dom cdm 5614  PosetRelcps 18462   TosetRel ctsr 18463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-tsr 18465

This theorem is referenced by:  tsrlemax  18484  ordtrest2lem  23111  ordthauslem  23291  ordthaus  23292