Theorem elinel1 3407

Description: Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
elinel1	⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵)

Proof of Theorem elinel1

Step	Hyp	Ref	Expression
1		elin 3404	. 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
2	1	simplbi 274	1 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2205   ∩ cin 3212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3219

This theorem is referenced by:  elin1d  3410  ressuppss  6456  fival  7259  fi0  7264  hashfibclem  11210  ballotfilem2  13149  ballotfilemfp1  13152  2idlval  14667  blbas  15315  blres  15316  pilem3  15665  uhgrspansubgrlem  16288  taupi  16876