Theorem elini 4099

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 4096	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
4	1, 2, 3	mpbir2an 724	1 ⊢ 𝐴 ∈ (𝐵 ∩ 𝐶)

This proof depends on syntax axioms:   ∈ wcel 2147   ∩ cin 4092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094

This theorem is used by:  isfin1-3  10503  setc2ohom  18320  isdrs2  18530  fpwipodrs  18764  0cmp  23767  comppfsc  23906  ptcmpfi  24187  alexsubALTlem2  24422  alexsubALTlem4  24424  ptcmp  24432  cnstrcvs  25517  cncvs  25521  recvs  25522  qcvs  25523  cnncvs  25535  ovolicc1  25892  ioorf  25949  zringpid  34187  corclrcl  44968  0pwfi  46350  sge0rnn0  47651  sge0reuz  47730  numtowerdt  48189  termc2  50984