Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elinti Structured version   Visualization version   GIF version

Theorem elinti 4797
 Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
elinti (𝐴 𝐵 → (𝐶𝐵𝐴𝐶))

Proof of Theorem elinti
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elintg 4796 . . 3 (𝐴 𝐵 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
2 eleq2 2873 . . . 4 (𝑥 = 𝐶 → (𝐴𝑥𝐴𝐶))
32rspccv 3558 . . 3 (∀𝑥𝐵 𝐴𝑥 → (𝐶𝐵𝐴𝐶))
41, 3syl6bi 254 . 2 (𝐴 𝐵 → (𝐴 𝐵 → (𝐶𝐵𝐴𝐶)))
54pm2.43i 52 1 (𝐴 𝐵 → (𝐶𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2083  ∀wral 3107  ∩ cint 4788 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-v 3442  df-int 4789 This theorem is referenced by:  inttsk  10049  subgint  18061  subrgint  19251  lssintcl  19430  ufinffr  22225  shintcli  28793  insiga  31009  intsal  42177
 Copyright terms: Public domain W3C validator