Theorem ssrin 3481

Description: Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

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

Proof of Theorem ssrin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . 4 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	anim1d 547	. . 3 ⊢ (A ⊆ B → ((x ∈ A ∧ x ∈ C) → (x ∈ B ∧ x ∈ C)))
3		elin 3220	. . 3 ⊢ (x ∈ (A ∩ C) ↔ (x ∈ A ∧ x ∈ C))
4		elin 3220	. . 3 ⊢ (x ∈ (B ∩ C) ↔ (x ∈ B ∧ x ∈ C))
5	2, 3, 4	3imtr4g 261	. 2 ⊢ (A ⊆ B → (x ∈ (A ∩ C) → x ∈ (B ∩ C)))
6	5	ssrdv 3279	1 ⊢ (A ⊆ B → (A ∩ C) ⊆ (B ∩ C))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∈ wcel 1710   ∩ cin 3209   ⊆ wss 3258

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is used by:  sslin  3482  ss2in  3483  ssdisj  3601  ssdifin0  3632  pw1ss  4170  ssres  4991