MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difeqri Structured version   Visualization version   GIF version

Theorem difeqri 4083
Description: Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypothesis
Ref Expression
difeqri.1 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)
Assertion
Ref Expression
difeqri (𝐴𝐵) = 𝐶
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem difeqri
StepHypRef Expression
1 eldif 3919 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 difeqri.1 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 274 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2733 1 (𝐴𝐵) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396   = wceq 1541  wcel 2106  cdif 3906
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3446  df-dif 3912
This theorem is referenced by:  difdif  4089  ddif  4095  dfss4  4217  difin  4220  difab  4259
  Copyright terms: Public domain W3C validator