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

Theorem difidALT 3436
Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3435. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
difidALT (𝐴𝐴) = ∅

Proof of Theorem difidALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfdif2 3083 . 2 (𝐴𝐴) = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
2 dfnul3 3370 . 2 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
31, 2eqtr4i 2164 1 (𝐴𝐴) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1332  wcel 1481  {crab 2421  cdif 3072  c0 3367
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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rab 2426  df-v 2691  df-dif 3077  df-nul 3368
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator