Theorem sotr3 31656

Description: Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)

Assertion

Ref	Expression
sotr3	|- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( ( X R Y /\ -. Z R Y ) -> X R Z ) )

Proof of Theorem sotr3

Step	Hyp	Ref	Expression
1		simp3 1063	. . . . . . 7 |- ( ( X e. A /\ Y e. A /\ Z e. A ) -> Z e. A )
2		simp2 1062	. . . . . . 7 |- ( ( X e. A /\ Y e. A /\ Z e. A ) -> Y e. A )
3	1, 2	jca 554	. . . . . 6 |- ( ( X e. A /\ Y e. A /\ Z e. A ) -> ( Z e. A /\ Y e. A ) )
4		sotric 5061	. . . . . 6 |- ( ( R Or A /\ ( Z e. A /\ Y e. A ) ) -> ( Z R Y <-> -. ( Z = Y \/ Y R Z ) ) )
5	3, 4	sylan2 491	. . . . 5 |- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( Z R Y <-> -. ( Z = Y \/ Y R Z ) ) )
6	5	con2bid 344	. . . 4 |- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( ( Z = Y \/ Y R Z ) <-> -. Z R Y ) )
7	6	adantr 481	. . 3 |- ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( ( Z = Y \/ Y R Z ) <-> -. Z R Y ) )
8		breq2 4657	. . . . . 6 |- ( Z = Y -> ( X R Z <-> X R Y ) )
9	8	biimprcd 240	. . . . 5 |- ( X R Y -> ( Z = Y -> X R Z ) )
10	9	adantl 482	. . . 4 |- ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( Z = Y -> X R Z ) )
11		sotr 5057	. . . . 5 |- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( ( X R Y /\ Y R Z ) -> X R Z ) )
12	11	expdimp 453	. . . 4 |- ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( Y R Z -> X R Z ) )
13	10, 12	jaod 395	. . 3 |- ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( ( Z = Y \/ Y R Z ) -> X R Z ) )
14	7, 13	sylbird 250	. 2 |- ( ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) /\ X R Y ) -> ( -. Z R Y -> X R Z ) )
15	14	expimpd 629	1 |- ( ( R Or A /\ ( X e. A /\ Y e. A /\ Z e. A ) ) -> ( ( X R Y /\ -. Z R Y ) -> X R Z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   Or wor 5034

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  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-po 5035  df-so 5036

This theorem is referenced by:  nosupbnd2  31862  sltletr  31881