Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssini Structured version   GIF version

Theorem ssini 3154
 Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
Hypotheses
Ref Expression
ssini.1 AB
ssini.2 A𝐶
Assertion
Ref Expression
ssini A ⊆ (B𝐶)

Proof of Theorem ssini
StepHypRef Expression
1 ssini.1 . . 3 AB
2 ssini.2 . . 3 A𝐶
31, 2pm3.2i 257 . 2 (AB A𝐶)
4 ssin 3153 . 2 ((AB A𝐶) ↔ A ⊆ (B𝐶))
53, 4mpbi 133 1 A ⊆ (B𝐶)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∩ 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:  inv1  3247
 Copyright terms: Public domain W3C validator