Theorem isso2i 4807

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 1842	. . . . 5 |- x = x
2	1	orci 391	. . . 4 |- ( x = x \/ x R x )
3		eleq1 2501	. . . . . . 7 |- ( y = x -> ( y e. A <-> x e. A ) )
4	3	anbi2d 708	. . . . . 6 |- ( y = x -> ( ( x e. A /\ y e. A ) <-> ( x e. A /\ x e. A ) ) )
5		equequ2 1851	. . . . . . . 8 |- ( y = x -> ( x = y <-> x = x ) )
6		breq1 4429	. . . . . . . 8 |- ( y = x -> ( y R x <-> x R x ) )
7	5, 6	orbi12d 714	. . . . . . 7 |- ( y = x -> ( ( x = y \/ y R x ) <-> ( x = x \/ x R x ) ) )
8		breq2 4430	. . . . . . . 8 |- ( y = x -> ( x R y <-> x R x ) )
9	8	notbid 295	. . . . . . 7 |- ( y = x -> ( -. x R y <-> -. x R x ) )
10	7, 9	bibi12d 322	. . . . . 6 |- ( y = x -> ( ( ( x = y \/ y R x ) <-> -. x R y ) <-> ( ( x = x \/ x R x ) <-> -. x R x ) ) )
11	4, 10	imbi12d 321	. . . . 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 330	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( x = y \/ y R x ) <-> -. x R y ) )
14	11, 13	chvarv 2070	. . . 4 |- ( ( x e. A /\ x e. A ) -> ( ( x = x \/ x R x ) <-> -. x R x ) )
15	2, 14	mpbii 214	. . 3 |- ( ( x e. A /\ x e. A ) -> -. x R x )
16	15	anidms 649	. 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 226	. . 3 |- ( ( x e. A /\ y e. A ) -> ( -. x R y -> ( x = y \/ y R x ) ) )
19		3orass 985	. . . 4 |- ( ( x R y \/ x = y \/ y R x ) <-> ( x R y \/ ( x = y \/ y R x ) ) )
20		df-or 371	. . . 4 |- ( ( x R y \/ ( x = y \/ y R x ) ) <-> ( -. x R y -> ( x = y \/ y R x ) ) )
21	19, 20	bitri 252	. . 3 |- ( ( x R y \/ x = y \/ y R x ) <-> ( -. x R y -> ( x = y \/ y R x ) ) )
22	18, 21	sylibr 215	. 2 |- ( ( x e. A /\ y e. A ) -> ( x R y \/ x = y \/ y R x ) )
23	16, 17, 22	issoi 4806	1 |- R Or A

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   \/ w3o 981   /\ w3a 982   e. wcel 1870   class class class wbr 4426   Or wor 4774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-po 4775  df-so 4776

This theorem is referenced by:  ltsonq  9393  ltsosr  9517  ltso  9713  xrltso  11440