Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  elin1d GIF version

Theorem elin1d 3296
 Description: Elementhood in the first set of an intersection - deduction version. (Contributed by Thierry Arnoux, 3-May-2020.)
Hypothesis
Ref Expression
elin1d.1 (𝜑𝑋 ∈ (𝐴𝐵))
Assertion
Ref Expression
elin1d (𝜑𝑋𝐴)

Proof of Theorem elin1d
StepHypRef Expression
1 elin1d.1 . 2 (𝜑𝑋 ∈ (𝐴𝐵))
2 elinel1 3293 . 2 (𝑋 ∈ (𝐴𝐵) → 𝑋𝐴)
31, 2syl 14 1 (𝜑𝑋𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 2128   ∩ cin 3101 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108 This theorem is referenced by:  fiuni  6915  explecnv  11384  restbasg  12528  txcnp  12631  blin2  12792  bj-charfun  13342
 Copyright terms: Public domain W3C validator