Theorem elin2 4198

Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)

Hypothesis

Ref	Expression
elin2.x	⊢ 𝑋 = (𝐵 ∩ 𝐶)

Assertion

Ref	Expression
elin2	⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

Proof of Theorem elin2

Step	Hyp	Ref	Expression
1		elin2.x	. . 3 ⊢ 𝑋 = (𝐵 ∩ 𝐶)
2	1	eleq2i 2826	. 2 ⊢ (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝐵 ∩ 𝐶))
3		elin 3965	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	2, 3	bitri 275	1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∩ cin 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956

This theorem is referenced by:  elin3  4201  opelres  5988  elpredgg  6314  fnres  6678  funfvima  7232  fnwelem  8117  ressuppssdif  8170  fz1isolem  14422  isabl  19652  isfld  20368  df2idl2  20860  2idlelb  20865  qus1  20872  qusrhm  20874  isidom  20922  lmres  22804  isnvc  24212  cvslvec  24641  cvsclm  24642  iscvs  24643  cvsi  24646  ishl  24879  ply1pid  25697  rplogsum  27030  sltres  27165  iscusgr  28675  isphg  30070  ishlo  30140  hhsscms  30531  mayete3i  30981  isogrp  32220  isofld  32420  bj-elid6  36051  bj-isrvec  36175  caures  36628  iscrngo  36864  fldcrngo  36872  isdmn  36922  isolat  38082  srhmsubclem1  46971  srhmsubc  46974  srhmsubcALTV  46992