Theorem tsrlin 17219

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 e. TosetRel /\ A e. X /\ B e. X ) -> ( A R B \/ B R A ) )

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 17218	. . . 4 |- ( R e. TosetRel <-> ( R e. PosetRel /\ A. x e. X A. y e. X ( x R y \/ y R x ) ) )
3	2	simprbi 480	. . 3 |- ( R e. TosetRel -> A. x e. X A. y e. X ( x R y \/ y R x ) )
4		breq1 4656	. . . . 5 |- ( x = A -> ( x R y <-> A R y ) )
5		breq2 4657	. . . . 5 |- ( x = A -> ( y R x <-> y R A ) )
6	4, 5	orbi12d 746	. . . 4 |- ( x = A -> ( ( x R y \/ y R x ) <-> ( A R y \/ y R A ) ) )
7		breq2 4657	. . . . 5 |- ( y = B -> ( A R y <-> A R B ) )
8		breq1 4656	. . . . 5 |- ( y = B -> ( y R A <-> B R A ) )
9	7, 8	orbi12d 746	. . . 4 |- ( y = B -> ( ( A R y \/ y R A ) <-> ( A R B \/ B R A ) ) )
10	6, 9	rspc2v 3322	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X ( x R y \/ y R x ) -> ( A R B \/ B R A ) ) )
11	3, 10	syl5com 31	. 2 |- ( R e. TosetRel -> ( ( A e. X /\ B e. X ) -> ( A R B \/ B R A ) ) )
12	11	3impib 1262	1 |- ( ( R e. TosetRel /\ A e. X /\ B e. X ) -> ( A R B \/ B R A ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   dom cdm 5114   PosetRelcps 17198   TosetRel ctsr 17199

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  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-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-tsr 17201

This theorem is referenced by:  tsrlemax  17220  ordtrest2lem  21007  ordthauslem  21187  ordthaus  21188