Theorem ssrin 1661

Description: Add right intersection to subclass relation.

Assertion

Ref	Expression
ssrin	⊢ (A ⊆ B → (A ∩ C) ⊆ (B ∩ C))

Proof of Theorem ssrin

Step	Hyp	Ref	Expression
1		id 9	. . . . 5 ⊢ ((x ∈ A → x ∈ B) → (x ∈ A → x ∈ B))
2	1	anim1d 432	. . . 4 ⊢ ((x ∈ A → x ∈ B) → ((x ∈ A ∧ x ∈ C) → (x ∈ B ∧ x ∈ C)))
3		elin 1635	. . . 4 ⊢ (x ∈ (A ∩ C) ↔ (x ∈ A ∧ x ∈ C))
4		elin 1635	. . . 4 ⊢ (x ∈ (B ∩ C) ↔ (x ∈ B ∧ x ∈ C))
5	2, 3, 4	3imtr4g 426	. . 3 ⊢ ((x ∈ A → x ∈ B) → (x ∈ (A ∩ C) → x ∈ (B ∩ C)))
6	5	19.20i 691	. 2 ⊢ (∀x(x ∈ A → x ∈ B) → ∀x(x ∈ (A ∩ C) → x ∈ (B ∩ C)))
7		dfss2 1497	. 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
8		dfss2 1497	. 2 ⊢ ((A ∩ C) ⊆ (B ∩ C) ↔ ∀x(x ∈ (A ∩ C) → x ∈ (B ∩ C)))
9	6, 7, 8	3imtr4 192	1 ⊢ (A ⊆ B → (A ∩ C) ⊆ (B ∩ C))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487

This theorem is referenced by:  sslin 1662  ss2in 1663  ssres 2589  sbthlem7 3355  phplem3 3405  orthin 5371  3oalem6 5557  mdbr2 5728  sumdmdlem 5786

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-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492

metamath.org