Theorem elini 4140

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 3906	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	1, 2, 3	mpbir2an 712	1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)

Syntax hints:   ∈ wcel 2114   ∩ cin 3889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-in 3897

This theorem is referenced by:  isfin1-3  10299  setc2ohom  18053  isdrs2  18263  fpwipodrs  18497  0cmp  23369  comppfsc  23507  ptcmpfi  23788  alexsubALTlem2  24023  alexsubALTlem4  24025  ptcmp  24033  cnstrcvs  25118  cncvs  25122  recvs  25123  qcvs  25124  cnncvs  25136  ovolicc1  25493  ioorf  25550  zringpid  33627  corclrcl  44152  0pwfi  45508  sge0rnn0  46814  sge0reuz  46893  nthrucw  47332  termc2  50005