Theorem elini 4154

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 3923	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	1, 2, 3	mpbir2an 723	1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)

Syntax hints:   ∈ wcel 2145   ∩ cin 3906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3914

This theorem is referenced by:  isfin1-3  10358  setc2ohom  18140  isdrs2  18350  fpwipodrs  18584  0cmp  23508  comppfsc  23646  ptcmpfi  23927  alexsubALTlem2  24162  alexsubALTlem4  24164  ptcmp  24172  cnstrcvs  25257  cncvs  25261  recvs  25262  qcvs  25263  cnncvs  25275  ovolicc1  25632  ioorf  25689  zringpid  33754  corclrcl  44290  0pwfi  45638  sge0rnn0  46941  sge0reuz  47020  nthrucw  47461  termc2  50148