Theorem tleile 29661

Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Hypotheses

Ref	Expression
tleile.b	|- B = ( Base ` K )
tleile.l	|- .<_ = ( le ` K )

Assertion

Ref	Expression
tleile	|- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .<_ Y \/ Y .<_ X ) )

Proof of Theorem tleile

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

Step	Hyp	Ref	Expression
1		simp2 1062	. 2 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> X e. B )
2		simp3 1063	. 2 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> Y e. B )
3		tleile.b	. . . . 5 |- B = ( Base ` K )
4		tleile.l	. . . . 5 |- .<_ = ( le ` K )
5	3, 4	istos 17035	. . . 4 |- ( K e. Toset <-> ( K e. Poset /\ A. x e. B A. y e. B ( x .<_ y \/ y .<_ x ) ) )
6	5	simprbi 480	. . 3 |- ( K e. Toset -> A. x e. B A. y e. B ( x .<_ y \/ y .<_ x ) )
7	6	3ad2ant1 1082	. 2 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> A. x e. B A. y e. B ( x .<_ y \/ y .<_ x ) )
8		breq1 4656	. . . 4 |- ( x = X -> ( x .<_ y <-> X .<_ y ) )
9		breq2 4657	. . . 4 |- ( x = X -> ( y .<_ x <-> y .<_ X ) )
10	8, 9	orbi12d 746	. . 3 |- ( x = X -> ( ( x .<_ y \/ y .<_ x ) <-> ( X .<_ y \/ y .<_ X ) ) )
11		breq2 4657	. . . 4 |- ( y = Y -> ( X .<_ y <-> X .<_ Y ) )
12		breq1 4656	. . . 4 |- ( y = Y -> ( y .<_ X <-> Y .<_ X ) )
13	11, 12	orbi12d 746	. . 3 |- ( y = Y -> ( ( X .<_ y \/ y .<_ X ) <-> ( X .<_ Y \/ Y .<_ X ) ) )
14	10, 13	rspc2va 3323	. 2 |- ( ( ( X e. B /\ Y e. B ) /\ A. x e. B A. y e. B ( x .<_ y \/ y .<_ x ) ) -> ( X .<_ Y \/ Y .<_ X ) )
15	1, 2, 7, 14	syl21anc 1325	1 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .<_ Y \/ Y .<_ X ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940  Tosetctos 17033

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-nul 4789

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-toset 17034

This theorem is referenced by:  tltnle  29662  odutos  29663  trleile  29666