Theorem soss 4042

Description: Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
soss	⊢ (A ⊆ B → (𝑅 Or B → 𝑅 Or A))

Proof of Theorem soss

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poss 4026	. . 3 ⊢ (A ⊆ B → (𝑅 Po B → 𝑅 Po A))
2		ssel 2933	. . . . . . . 8 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
3		ssel 2933	. . . . . . . 8 ⊢ (A ⊆ B → (y ∈ A → y ∈ B))
4		ssel 2933	. . . . . . . 8 ⊢ (A ⊆ B → (z ∈ A → z ∈ B))
5	2, 3, 4	3anim123d 1213	. . . . . . 7 ⊢ (A ⊆ B → ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x ∈ B ∧ y ∈ B ∧ z ∈ B)))
6	5	imim1d 69	. . . . . 6 ⊢ (A ⊆ B → (((x ∈ B ∧ y ∈ B ∧ z ∈ B) → (x𝑅y → (x𝑅z ∨ z𝑅y))) → ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x𝑅y → (x𝑅z ∨ z𝑅y)))))
7	6	2alimdv 1758	. . . . 5 ⊢ (A ⊆ B → (∀y∀z((x ∈ B ∧ y ∈ B ∧ z ∈ B) → (x𝑅y → (x𝑅z ∨ z𝑅y))) → ∀y∀z((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x𝑅y → (x𝑅z ∨ z𝑅y)))))
8	7	alimdv 1756	. . . 4 ⊢ (A ⊆ B → (∀x∀y∀z((x ∈ B ∧ y ∈ B ∧ z ∈ B) → (x𝑅y → (x𝑅z ∨ z𝑅y))) → ∀x∀y∀z((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x𝑅y → (x𝑅z ∨ z𝑅y)))))
9		r3al 2360	. . . 4 ⊢ (∀x ∈ B ∀y ∈ B ∀z ∈ B (x𝑅y → (x𝑅z ∨ z𝑅y)) ↔ ∀x∀y∀z((x ∈ B ∧ y ∈ B ∧ z ∈ B) → (x𝑅y → (x𝑅z ∨ z𝑅y))))
10		r3al 2360	. . . 4 ⊢ (∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y)) ↔ ∀x∀y∀z((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x𝑅y → (x𝑅z ∨ z𝑅y))))
11	8, 9, 10	3imtr4g 194	. . 3 ⊢ (A ⊆ B → (∀x ∈ B ∀y ∈ B ∀z ∈ B (x𝑅y → (x𝑅z ∨ z𝑅y)) → ∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y))))
12	1, 11	anim12d 318	. 2 ⊢ (A ⊆ B → ((𝑅 Po B ∧ ∀x ∈ B ∀y ∈ B ∀z ∈ B (x𝑅y → (x𝑅z ∨ z𝑅y))) → (𝑅 Po A ∧ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y)))))
13		df-iso 4025	. 2 ⊢ (𝑅 Or B ↔ (𝑅 Po B ∧ ∀x ∈ B ∀y ∈ B ∀z ∈ B (x𝑅y → (x𝑅z ∨ z𝑅y))))
14		df-iso 4025	. 2 ⊢ (𝑅 Or A ↔ (𝑅 Po A ∧ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y))))
15	12, 13, 14	3imtr4g 194	1 ⊢ (A ⊆ B → (𝑅 Or B → 𝑅 Or A))

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 628   ∧ w3a 884  ∀wal 1240   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911   class class class wbr 3755   Po wpo 4022   Or wor 4023

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-in 2918  df-ss 2925  df-po 4024  df-iso 4025

This theorem is referenced by:  soeq2  4044