Theorem soss 2852

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 2841	. . 3 |- (A (_ B -> (R Po B -> R Po A))
2		ssel 2063	. . . . . . . 8 |- (A (_ B -> (x e. A -> x e. B))
3		ssel 2063	. . . . . . . 8 |- (A (_ B -> (y e. A -> y e. B))
4	2, 3	anim12d 558	. . . . . . 7 |- (A (_ B -> ((x e. A /\ y e. A) -> (x e. B /\ y e. B)))
5	4	imim1d 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 1289	. . . . 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 1289	. . . 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 1676	. . . 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 1676	. . . 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 553	. . 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 558	. 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 2850	. 2 |- (R Or B <-> (R Po B /\ A.x e. B A.y e. B (xRy \/ x = y \/ yRx)))
13		df-so 2850	. 2 |- (R Or A <-> (R Po A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
14	11, 12, 13	3imtr4g 553	1 |- (A (_ B -> (R Or B -> R Or A))

Syntax hints:   -> wi 3   /\ wa 223   \/ w3o 774  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047   class class class wbr 2619   Po wpo 2838   Or wor 2839

This theorem is referenced by:  soeq2 2854  wess 2936

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-in 2051  df-ss 2053  df-po 2840  df-so 2850

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