Theorem elini 4162

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 3930	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	1, 2, 3	mpbir2an 711	1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)

Syntax hints:   ∈ wcel 2109   ∩ cin 3913

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 3449  df-in 3921

This theorem is referenced by:  isfin1-3  10339  setc2ohom  18057  isdrs2  18267  fpwipodrs  18499  0cmp  23281  comppfsc  23419  ptcmpfi  23700  alexsubALTlem2  23935  alexsubALTlem4  23937  ptcmp  23945  cnstrcvs  25041  cncvs  25045  recvs  25046  qcvs  25047  cnncvs  25059  ovolicc1  25417  ioorf  25474  zringpid  33523  corclrcl  43696  0pwfi  45053  sge0rnn0  46366  sge0reuz  46445  termc2  49507