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Theorem difid 3401
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. (Contributed by NM, 22-Apr-2004.)
Assertion
Ref Expression
difid (𝐴𝐴) = ∅

Proof of Theorem difid
StepHypRef Expression
1 ssid 3087 . 2 𝐴𝐴
2 ssdif0im 3397 . 2 (𝐴𝐴 → (𝐴𝐴) = ∅)
31, 2ax-mp 5 1 (𝐴𝐴) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1316  cdif 3038  wss 3041  c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by:  dif0  3403  difun2  3412  diftpsn3  3631  2oconcl  6304  ismkvnex  6997  topcld  12205
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