Theorem elini 4199

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 3967	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	1, 2, 3	mpbir2an 711	1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)

Syntax hints:   ∈ wcel 2108   ∩ cin 3950

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958

This theorem is referenced by:  isfin1-3  10426  setc2ohom  18140  isdrs2  18352  fpwipodrs  18585  0cmp  23402  comppfsc  23540  ptcmpfi  23821  alexsubALTlem2  24056  alexsubALTlem4  24058  ptcmp  24066  cnstrcvs  25174  cncvs  25178  recvs  25179  recvsOLD  25180  qcvs  25181  cnncvs  25193  ovolicc1  25551  ioorf  25608  zringpid  33580  corclrcl  43720  0pwfi  45064  sge0rnn0  46383  sge0reuz  46462  termc2  49148