Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elin2 Structured version   Visualization version   GIF version

Theorem elin2 3763
 Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypothesis
Ref Expression
elin2.x 𝑋 = (𝐵𝐶)
Assertion
Ref Expression
elin2 (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶))

Proof of Theorem elin2
StepHypRef Expression
1 elin2.x . . 3 𝑋 = (𝐵𝐶)
21eleq2i 2680 . 2 (𝐴𝑋𝐴 ∈ (𝐵𝐶))
3 elin 3758 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
42, 3bitri 263 1 (𝐴𝑋 ↔ (𝐴𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547 This theorem is referenced by:  elin3  3766  elpredim  5609  elpred  5610  elpredg  5611  fnres  5921  funfvima  6396  fnwelem  7179  ressuppssdif  7203  fz1isolem  13102  isabl  18020  isfld  18579  2idlcpbl  19055  qus1  19056  qusrhm  19058  isidom  19125  lmres  20914  isnvc  22309  cvslvec  22733  cvsclm  22734  iscvs  22735  ishl  22966  ply1pid  23743  rplogsum  25016  isphg  27056  ishlo  27127  hhsscms  27520  mayete3i  27971  isogrp  29033  isofld  29133  sltres  31061  nofulllem5  31105  caures  32726  iscrngo  32965  fldcrng  32973  isdmn  33023  isolat  33517  srhmsubclem1  41865  srhmsubc  41868  srhmsubcALTVlem1  41884  srhmsubcALTV  41887
 Copyright terms: Public domain W3C validator