Theorem isso2i 4669

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 1734	. . . . 5 |- x = x
2	1	orci 390	. . . 4 |- ( x = x \/ x R x )
3		eleq1 2501	. . . . . . 7 |- ( y = x -> ( y e. A <-> x e. A ) )
4	3	anbi2d 698	. . . . . 6 |- ( y = x -> ( ( x e. A /\ y e. A ) <-> ( x e. A /\ x e. A ) ) )
5		equequ2 1742	. . . . . . . 8 |- ( y = x -> ( x = y <-> x = x ) )
6		breq1 4292	. . . . . . . 8 |- ( y = x -> ( y R x <-> x R x ) )
7	5, 6	orbi12d 704	. . . . . . 7 |- ( y = x -> ( ( x = y \/ y R x ) <-> ( x = x \/ x R x ) ) )
8		breq2 4293	. . . . . . . 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 1963	. . . 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 640	. 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 963	. . . 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 4668	1 |- R Or A

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/ w3o 959   /\ w3a 960   e. wcel 1761   class class class wbr 4289   Or wor 4636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-po 4637  df-so 4638

This theorem is referenced by:  ltsonq  9134  ltsosr  9257  ltso  9451  xrltso  11114