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Theorem difid 3319
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 2991 . 2 𝐴𝐴
2 ssdif0im 3313 . 2 (𝐴𝐴 → (𝐴𝐴) = ∅)
31, 2ax-mp 7 1 (𝐴𝐴) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cdif 2941  wss 2944  c0 3251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2947  df-in 2951  df-ss 2958  df-nul 3252
This theorem is referenced by:  dif0  3321  difun2  3329  diftpsn3  3532  2oconcl  6052
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