Theorem elini 4152

Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
elini.1	⊢ 𝐴 ∈ 𝐵
elini.2	⊢ 𝐴 ∈ 𝐶

Assertion

Ref	Expression
elini	⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)

Proof of Theorem elini

Step	Hyp	Ref	Expression
1		elini.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		elini.2	. 2 ⊢ 𝐴 ∈ 𝐶
3		elin 3921	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	1, 2, 3	mpbir2an 711	1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)

Syntax hints:   ∈ wcel 2109   ∩ cin 3904

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 3440  df-in 3912

This theorem is referenced by:  isfin1-3  10299  setc2ohom  18020  isdrs2  18230  fpwipodrs  18464  0cmp  23297  comppfsc  23435  ptcmpfi  23716  alexsubALTlem2  23951  alexsubALTlem4  23953  ptcmp  23961  cnstrcvs  25057  cncvs  25061  recvs  25062  qcvs  25063  cnncvs  25075  ovolicc1  25433  ioorf  25490  zringpid  33499  corclrcl  43680  0pwfi  45037  sge0rnn0  46350  sge0reuz  46429  termc2  49504