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Theorem difidALT 3492
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 3491. (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 3137 . 2 (𝐴𝐴) = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
2 dfnul3 3425 . 2 ∅ = {𝑥𝐴 ∣ ¬ 𝑥𝐴}
31, 2eqtr4i 2201 1 (𝐴𝐴) = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1353  wcel 2148  {crab 2459  cdif 3126  c0 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2739  df-dif 3131  df-nul 3423
This theorem is referenced by: (None)
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