Theorem poss 4026

Description: Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
poss	⊢ (A ⊆ B → (𝑅 Po B → 𝑅 Po A))

Proof of Theorem poss

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

Step	Hyp	Ref	Expression
1		ssralv 2998	. . 3 ⊢ (A ⊆ B → (∀x ∈ B ∀y ∈ B ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) → ∀x ∈ A ∀y ∈ B ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z))))
2		ssralv 2998	. . . . 5 ⊢ (A ⊆ B → (∀y ∈ B ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) → ∀y ∈ A ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z))))
3		ssralv 2998	. . . . . 6 ⊢ (A ⊆ B → (∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) → ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z))))
4	3	ralimdv 2382	. . . . 5 ⊢ (A ⊆ B → (∀y ∈ A ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) → ∀y ∈ A ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z))))
5	2, 4	syld 40	. . . 4 ⊢ (A ⊆ B → (∀y ∈ B ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) → ∀y ∈ A ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z))))
6	5	ralimdv 2382	. . 3 ⊢ (A ⊆ B → (∀x ∈ A ∀y ∈ B ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) → ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z))))
7	1, 6	syld 40	. 2 ⊢ (A ⊆ B → (∀x ∈ B ∀y ∈ B ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)) → ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z))))
8		df-po 4024	. 2 ⊢ (𝑅 Po B ↔ ∀x ∈ B ∀y ∈ B ∀z ∈ B (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
9		df-po 4024	. 2 ⊢ (𝑅 Po A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ x𝑅x ∧ ((x𝑅y ∧ y𝑅z) → x𝑅z)))
10	7, 8, 9	3imtr4g 194	1 ⊢ (A ⊆ B → (𝑅 Po B → 𝑅 Po A))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wral 2300   ⊆ wss 2911   class class class wbr 3755   Po wpo 4022

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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925  df-po 4024

This theorem is referenced by:  poeq2  4028  soss  4042