Theorem isso2i 4786

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 1854	. . . . 5 |- x = x
2	1	orci 392	. . . 4 |- ( x = x \/ x R x )
3		eleq1 2516	. . . . . . 7 |- ( y = x -> ( y e. A <-> x e. A ) )
4	3	anbi2d 709	. . . . . 6 |- ( y = x -> ( ( x e. A /\ y e. A ) <-> ( x e. A /\ x e. A ) ) )
5		equequ2 1867	. . . . . . . 8 |- ( y = x -> ( x = y <-> x = x ) )
6		breq1 4404	. . . . . . . 8 |- ( y = x -> ( y R x <-> x R x ) )
7	5, 6	orbi12d 715	. . . . . . 7 |- ( y = x -> ( ( x = y \/ y R x ) <-> ( x = x \/ x R x ) ) )
8		breq2 4405	. . . . . . . 8 |- ( y = x -> ( x R y <-> x R x ) )
9	8	notbid 296	. . . . . . 7 |- ( y = x -> ( -. x R y <-> -. x R x ) )
10	7, 9	bibi12d 323	. . . . . 6 |- ( y = x -> ( ( ( x = y \/ y R x ) <-> -. x R y ) <-> ( ( x = x \/ x R x ) <-> -. x R x ) ) )
11	4, 10	imbi12d 322	. . . . 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 331	. . . . 5 |- ( ( x e. A /\ y e. A ) -> ( ( x = y \/ y R x ) <-> -. x R y ) )
14	11, 13	chvarv 2106	. . . 4 |- ( ( x e. A /\ x e. A ) -> ( ( x = x \/ x R x ) <-> -. x R x ) )
15	2, 14	mpbii 215	. . 3 |- ( ( x e. A /\ x e. A ) -> -. x R x )
16	15	anidms 650	. 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 227	. . 3 |- ( ( x e. A /\ y e. A ) -> ( -. x R y -> ( x = y \/ y R x ) ) )
19		3orass 987	. . . 4 |- ( ( x R y \/ x = y \/ y R x ) <-> ( x R y \/ ( x = y \/ y R x ) ) )
20		df-or 372	. . . 4 |- ( ( x R y \/ ( x = y \/ y R x ) ) <-> ( -. x R y -> ( x = y \/ y R x ) ) )
21	19, 20	bitri 253	. . 3 |- ( ( x R y \/ x = y \/ y R x ) <-> ( -. x R y -> ( x = y \/ y R x ) ) )
22	18, 21	sylibr 216	. 2 |- ( ( x e. A /\ y e. A ) -> ( x R y \/ x = y \/ y R x ) )
23	16, 17, 22	issoi 4785	1 |- R Or A

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   \/ w3o 983   /\ w3a 984   e. wcel 1886   class class class wbr 4401   Or wor 4753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-po 4754  df-so 4755

This theorem is referenced by:  ltsonq  9391  ltsosr  9515  ltso  9711  xrltso  11437