Theorem soss 2140

Description: Subset theorem for the strict ordering predicate.

Assertion

Ref	Expression
soss	|- (A (_ B -> (R Or B -> R Or A))

Proof of Theorem soss

Step	Hyp	Ref	Expression
1		poss 2129	. . 3 |- (A (_ B -> (R Po B -> R Po A))
2		ssel 1502	. . . . . . . 8 |- (A (_ B -> (x e. A -> x e. B))
3		ssel 1502	. . . . . . . 8 |- (A (_ B -> (y e. A -> y e. B))
4	2, 3	anim12d 431	. . . . . . 7 |- (A (_ B -> ((x e. A /\ y e. A) -> (x e. B /\ y e. B)))
5	4	syl4d 28	. . . . . 6 |- (A (_ B -> (((x e. B /\ y e. B) -> (xRy \/ x = y \/ yRx)) -> ((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx))))
6	5	19.20dv 946	. . . . 5 |- (A (_ B -> (A.y((x e. B /\ y e. B) -> (xRy \/ x = y \/ yRx)) -> A.y((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx))))
7	6	19.20dv 946	. . . 4 |- (A (_ B -> (A.xA.y((x e. B /\ y e. B) -> (xRy \/ x = y \/ yRx)) -> A.xA.y((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx))))
8		r2al 1231	. . . 4 |- (A.x e. B A.y e. B (xRy \/ x = y \/ yRx) <-> A.xA.y((x e. B /\ y e. B) -> (xRy \/ x = y \/ yRx)))
9		r2al 1231	. . . 4 |- (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.xA.y((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)))
10	7, 8, 9	3imtr4g 426	. . 3 |- (A (_ B -> (A.x e. B A.y e. B (xRy \/ x = y \/ yRx) -> A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
11	1, 10	anim12d 431	. 2 |- (A (_ B -> ((R Po B /\ A.x e. B A.y e. B (xRy \/ x = y \/ yRx)) -> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx))))
12		df-so 2138	. 2 |- (R Or B <-> (R Po B /\ A.x e. B A.y e. B (xRy \/ x = y \/ yRx)))
13		df-so 2138	. 2 |- (R Or A <-> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
14	11, 12, 13	3imtr4g 426	1 |- (A (_ B -> (R Or B -> R Or A))

Syntax hints:   -> wi 2   /\ wa 196   \/ w3o 580  A.wal 672   = weq 797   e. wcel 1092  A.wral 1201   (_ wss 1487   class class class wbr 2054   Po wpo 2058   Or wor 2059

This theorem is referenced by:  soeq2 2142  wess 2188

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-in 1491  df-ss 1492  df-po 2128  df-so 2138

metamath.org