Theorem ssrin 3156

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 2933	. . . 4 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	anim1d 319	. . 3 ⊢ (A ⊆ B → ((x ∈ A ∧ x ∈ 𝐶) → (x ∈ B ∧ x ∈ 𝐶)))
3		elin 3120	. . 3 ⊢ (x ∈ (A ∩ 𝐶) ↔ (x ∈ A ∧ x ∈ 𝐶))
4		elin 3120	. . 3 ⊢ (x ∈ (B ∩ 𝐶) ↔ (x ∈ B ∧ x ∈ 𝐶))
5	2, 3, 4	3imtr4g 194	. 2 ⊢ (A ⊆ B → (x ∈ (A ∩ 𝐶) → x ∈ (B ∩ 𝐶)))
6	5	ssrdv 2945	1 ⊢ (A ⊆ B → (A ∩ 𝐶) ⊆ (B ∩ 𝐶))

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390   ∩ cin 2910   ⊆ wss 2911

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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925

This theorem is referenced by:  sslin  3157  ss2in  3158  ssdisj  3271  ssdifin0  3298  ssres  4580