Theorem isso2i 4673

Description: Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
isso2i.1	|- ( ( x e. A /\ y e. A ) -> ( x R y <-> -. ( x = y \/ y R x ) ) )
isso2i.2	|- ( ( x e. A /\ y e. A /\ z e. A ) -> ( ( x R y /\ y R z ) -> x R z ) )

Assertion

Ref	Expression
isso2i	|- R Or A

Distinct variable groups:   x, y, z, R   x, A, y, z

Proof of Theorem isso2i

Step	Hyp	Ref	Expression
1		equid 1729	. . . . 5 |- x = x
2	1	orci 390	. . . 4 |- ( x = x \/ x R x )
3		eleq1 2503	. . . . . . 7 |- ( y = x -> ( y e. A <-> x e. A ) )
4	3	anbi2d 703	. . . . . 6 |- ( y = x -> ( ( x e. A /\ y e. A ) <-> ( x e. A /\ x e. A ) ) )
5		equequ2 1737	. . . . . . . 8 |- ( y = x -> ( x = y <-> x = x ) )
6		breq1 4295	. . . . . . . 8 |- ( y = x -> ( y R x <-> x R x ) )
7	5, 6	orbi12d 709	. . . . . . 7 |- ( y = x -> ( ( x = y \/ y R x ) <-> ( x = x \/ x R x ) ) )
8		breq2 4296	. . . . . . . 8 |- ( y = x -> ( x R y <-> x R x ) )
9	8	notbid 294	. . . . . . 7 |- ( y = x -> ( -. x R y <-> -. x R x ) )
10	7, 9	bibi12d 321	. . . . . 6 |- ( y = x -> ( ( ( x = y \/ y R x ) <-> -. x R y ) <-> ( ( x = x \/ x R x ) <-> -. x R x ) ) )
11	4, 10	imbi12d 320	. . . . 5 |- ( y = x -> ( ( ( x e. A /\ y e. A ) -> ( ( x = y \/ y R x ) <-> -. x R y ) ) <-> ( ( x e. A /\ x e. A ) -> ( ( x = x \/ x R x ) <-> -. x R x ) ) ) )
12		isso2i.1	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( x R y <-> -. ( x = y \/ y R x ) ) )
13	12	con2bid 329	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( x = y \/ y R x ) <-> -. x R y ) )
14	11, 13	chvarv 1958	. . . 4 |- ( ( x e. A /\ x e. A ) -> ( ( x = x \/ x R x ) <-> -. x R x ) )
15	2, 14	mpbii 211	. . 3 |- ( ( x e. A /\ x e. A ) -> -. x R x )
16	15	anidms 645	. 2 |- ( x e. A -> -. x R x )
17		isso2i.2	. 2 |- ( ( x e. A /\ y e. A /\ z e. A ) -> ( ( x R y /\ y R z ) -> x R z ) )
18	13	biimprd 223	. . 3 |- ( ( x e. A /\ y e. A ) -> ( -. x R y -> ( x = y \/ y R x ) ) )
19		3orass 968	. . . 4 |- ( ( x R y \/ x = y \/ y R x ) <-> ( x R y \/ ( x = y \/ y R x ) ) )
20		df-or 370	. . . 4 |- ( ( x R y \/ ( x = y \/ y R x ) ) <-> ( -. x R y -> ( x = y \/ y R x ) ) )
21	19, 20	bitri 249	. . 3 |- ( ( x R y \/ x = y \/ y R x ) <-> ( -. x R y -> ( x = y \/ y R x ) ) )
22	18, 21	sylibr 212	. 2 |- ( ( x e. A /\ y e. A ) -> ( x R y \/ x = y \/ y R x ) )
23	16, 17, 22	issoi 4672	1 |- R Or A

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/ w3o 964   /\ w3a 965   e. wcel 1756   class class class wbr 4292   Or wor 4640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-po 4641  df-so 4642

This theorem is referenced by:  ltsonq  9138  ltsosr  9261  ltso  9455  xrltso  11118