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 ∈ A → x ∈ B))
3		ssel 1502	. . . . . . . 8 ⊢ (A ⊆ B → (y ∈ A → y ∈ B))
4	2, 3	anim12d 431	. . . . . . 7 ⊢ (A ⊆ B → ((x ∈ A ∧ y ∈ A) → (x ∈ B ∧ y ∈ B)))
5	4	syl4d 28	. . . . . 6 ⊢ (A ⊆ B → (((x ∈ B ∧ y ∈ B) → (xRy ∨ x = y ∨ yRx)) → ((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx))))
6	5	19.20dv 946	. . . . 5 ⊢ (A ⊆ B → (∀y((x ∈ B ∧ y ∈ B) → (xRy ∨ x = y ∨ yRx)) → ∀y((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx))))
7	6	19.20dv 946	. . . 4 ⊢ (A ⊆ B → (∀x∀y((x ∈ B ∧ y ∈ B) → (xRy ∨ x = y ∨ yRx)) → ∀x∀y((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx))))
8		r2al 1231	. . . 4 ⊢ (∀x ∈ B ∀y ∈ B (xRy ∨ x = y ∨ yRx) ↔ ∀x∀y((x ∈ B ∧ y ∈ B) → (xRy ∨ x = y ∨ yRx)))
9		r2al 1231	. . . 4 ⊢ (∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx) ↔ ∀x∀y((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx)))
10	7, 8, 9	3imtr4g 426	. . 3 ⊢ (A ⊆ B → (∀x ∈ B ∀y ∈ B (xRy ∨ x = y ∨ yRx) → ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)))
11	1, 10	anim12d 431	. 2 ⊢ (A ⊆ B → ((R Po B ∧ ∀x ∈ B ∀y ∈ B (xRy ∨ x = y ∨ yRx)) → (R Po A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx))))
12		df-so 2138	. 2 ⊢ (R Or B ↔ (R Po B ∧ ∀x ∈ B ∀y ∈ B (xRy ∨ x = y ∨ yRx)))
13		df-so 2138	. 2 ⊢ (R Or A ↔ (R Po A ∧ ∀x ∈ A ∀y ∈ 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  ∀wal 672   = weq 797   ∈ wcel 1092  ∀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