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Theorem difidALT 3287
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 3286. (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 2920 . 2 (AA) = {x A ∣ ¬ x A}
2 dfnul3 3221 . 2 ∅ = {x A ∣ ¬ x A}
31, 2eqtr4i 2060 1 (AA) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1242   wcel 1390  {crab 2304  cdif 2908  c0 3218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-nul 3219
This theorem is referenced by: (None)
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