Theorem elin2 4162

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 2820	. 2 ⊢ (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝐵 ∩ 𝐶))
3		elin 3927	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	2, 3	bitri 275	1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-in 3918

This theorem is referenced by:  elin3  4165  opelres  5945  elpredgg  6275  fnres  6627  funfvima  7186  fnwelem  8087  ressuppssdif  8141  fz1isolem  14402  isabl  19698  isogrp  20038  srhmsubclem1  20597  srhmsubc  20600  isidom  20645  isfld  20660  isofld  20784  2idlelb  21195  qus1  21216  qusrhm  21218  lmres  23220  isnvc  24616  cvslvec  25058  cvsclm  25059  iscvs  25060  cvsi  25063  ishl  25295  ply1pid  26121  rplogsum  27471  sltres  27607  iscusgr  29398  isphg  30796  ishlo  30866  hhsscms  31257  mayete3i  31707  bj-elid6  37151  bj-isrvec  37275  caures  37747  iscrngo  37983  fldcrngo  37991  isdmn  38041  isolat  39198  srhmsubcALTV  48306