Theorem trleile 29666

Description: In a Toset, two elements must compare. (Contributed by Thierry Arnoux, 12-Sep-2018.)

Hypotheses

Ref	Expression
trleile.b	|- B = ( Base ` K )
trleile.l	|- .<_ = ( ( le ` K ) i^i ( B X. B ) )

Assertion

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

Proof of Theorem trleile

Step	Hyp	Ref	Expression
1		trleile.b	. . . 4 |- B = ( Base ` K )
2		eqid 2622	. . . 4 |- ( le ` K ) = ( le ` K )
3	1, 2	tleile 29661	. . 3 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X ( le ` K ) Y \/ Y ( le ` K ) X ) )
4		3simpc 1060	. . . . . 6 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X e. B /\ Y e. B ) )
5		brxp 5147	. . . . . 6 |- ( X ( B X. B ) Y <-> ( X e. B /\ Y e. B ) )
6	4, 5	sylibr 224	. . . . 5 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> X ( B X. B ) Y )
7		brin 4704	. . . . . 6 |- ( X ( ( le ` K ) i^i ( B X. B ) ) Y <-> ( X ( le ` K ) Y /\ X ( B X. B ) Y ) )
8	7	rbaib 947	. . . . 5 |- ( X ( B X. B ) Y -> ( X ( ( le ` K ) i^i ( B X. B ) ) Y <-> X ( le ` K ) Y ) )
9	6, 8	syl 17	. . . 4 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X ( ( le ` K ) i^i ( B X. B ) ) Y <-> X ( le ` K ) Y ) )
10	4	ancomd 467	. . . . . 6 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( Y e. B /\ X e. B ) )
11		brxp 5147	. . . . . 6 |- ( Y ( B X. B ) X <-> ( Y e. B /\ X e. B ) )
12	10, 11	sylibr 224	. . . . 5 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> Y ( B X. B ) X )
13		brin 4704	. . . . . 6 |- ( Y ( ( le ` K ) i^i ( B X. B ) ) X <-> ( Y ( le ` K ) X /\ Y ( B X. B ) X ) )
14	13	rbaib 947	. . . . 5 |- ( Y ( B X. B ) X -> ( Y ( ( le ` K ) i^i ( B X. B ) ) X <-> Y ( le ` K ) X ) )
15	12, 14	syl 17	. . . 4 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( Y ( ( le ` K ) i^i ( B X. B ) ) X <-> Y ( le ` K ) X ) )
16	9, 15	orbi12d 746	. . 3 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( ( X ( ( le ` K ) i^i ( B X. B ) ) Y \/ Y ( ( le ` K ) i^i ( B X. B ) ) X ) <-> ( X ( le ` K ) Y \/ Y ( le ` K ) X ) ) )
17	3, 16	mpbird 247	. 2 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X ( ( le ` K ) i^i ( B X. B ) ) Y \/ Y ( ( le ` K ) i^i ( B X. B ) ) X ) )
18		trleile.l	. . . 4 |- .<_ = ( ( le ` K ) i^i ( B X. B ) )
19	18	breqi 4659	. . 3 |- ( X .<_ Y <-> X ( ( le ` K ) i^i ( B X. B ) ) Y )
20	18	breqi 4659	. . 3 |- ( Y .<_ X <-> Y ( ( le ` K ) i^i ( B X. B ) ) X )
21	19, 20	orbi12i 543	. 2 |- ( ( X .<_ Y \/ Y .<_ X ) <-> ( X ( ( le ` K ) i^i ( B X. B ) ) Y \/ Y ( ( le ` K ) i^i ( B X. B ) ) X ) )
22	17, 21	sylibr 224	1 |- ( ( K e. Toset /\ X e. B /\ Y e. B ) -> ( X .<_ Y \/ Y .<_ X ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   class class class wbr 4653   X. cxp 5112   ` cfv 5888   Basecbs 15857   lecple 15948  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-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-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-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896  df-toset 17034

This theorem is referenced by:  ordtrest2NEWlem  29968  ordtconnlem1  29970