Theorem elin2 3952

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 2842	. 2 ⊢ (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ (𝐵 ∩ 𝐶))
3		elin 3947	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	2, 3	bitri 264	1 ⊢ (𝐴 ∈ 𝑋 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))

Syntax hints:   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ∩ cin 3722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730

This theorem is referenced by:  elin3  3955  elpredim  5834  elpred  5835  elpredg  5836  fnres  6146  funfvima  6638  fnwelem  7447  ressuppssdif  7471  fz1isolem  13447  isabl  18404  isfld  18966  2idlcpbl  19449  qus1  19450  qusrhm  19452  isidom  19519  lmres  21325  isnvc  22719  cvslvec  23144  cvsclm  23145  iscvs  23146  ishl  23377  ply1pid  24159  rplogsum  25437  iscusgr  26549  isphg  28012  ishlo  28083  hhsscms  28476  mayete3i  28927  isogrp  30042  isofld  30142  sltres  32152  caures  33888  iscrngo  34127  fldcrng  34135  isdmn  34185  opelresALTV  34374  isolat  35021  srhmsubclem1  42598  srhmsubc  42601  srhmsubcALTV  42619