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

Theorem difeqri 3992
 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 3840 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝐵))
2 difeqri.1 . . 3 ((𝑥𝐴 ∧ ¬ 𝑥𝐵) ↔ 𝑥𝐶)
31, 2bitri 267 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥𝐶)
43eqriv 2776 1 (𝐴𝐵) = 𝐶
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ∖ cdif 3827 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-v 3418  df-dif 3833 This theorem is referenced by:  difdif  3998  ddif  4004  dfss4  4123  difin  4126  difab  4160
 Copyright terms: Public domain W3C validator