Theorem tsrlin 18512

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 18511	. . . 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 5102	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
5		breq2 5103	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝐴))
6	4, 5	orbi12d 919	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴)))
7		breq2 5103	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
8		breq1 5102	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝑅𝐴 ↔ 𝐵𝑅𝐴))
9	7, 8	orbi12d 919	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ∨ 𝑦𝑅𝐴) ↔ (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴)))
10	6, 9	rspc2v 3588	. . 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 1117	1 ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5099  dom cdm 5625  PosetRelcps 18491   TosetRel ctsr 18492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-tsr 18494

This theorem is referenced by:  tsrlemax  18513  ordtrest2lem  23151  ordthauslem  23331  ordthaus  23332