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Theorem difidALT 3270
 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 3269. (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 2903 . 2 (AA) = {x A ∣ ¬ x A}
2 dfnul3 3204 . 2 ∅ = {x A ∣ ¬ x A}
31, 2eqtr4i 2045 1 (AA) = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1228   ∈ wcel 1374  {crab 2288   ∖ cdif 2891  ∅c0 3201 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537  df-dif 2897  df-nul 3202 This theorem is referenced by: (None)
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