Theorem tsrlin 18624

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 18623	. . . 4 ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
3	2	simprbi 495	. . 3 ⊢ (𝑅 ∈ TosetRel → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))
4		breq1 5157	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
5		breq2 5158	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
6	4, 5	orbi12d 916	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴)))
7		breq2 5158	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
8		breq1 5157	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
9	7, 8	orbi12d 916	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴)))
10	6, 9	rspc2v 3631	. . 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 1113	1 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   = wceq 1534   ∈ wcel 2100  ∀wral 3054   class class class wbr 5154  dom cdm 5684  PosetRelcps 18603   TosetRel ctsr 18604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3961  df-un 3963  df-ss 3975  df-nul 4334  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5155  df-opab 5217  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-tsr 18606

This theorem is referenced by:  tsrlemax  18625  ordtrest2lem  23199  ordthauslem  23379  ordthaus  23380