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Theorem difexg 3972
Description: Existence of a difference. (Contributed by NM, 26-May-1998.)
Assertion
Ref Expression
difexg (𝐴𝑉 → (𝐴𝐵) ∈ V)

Proof of Theorem difexg
StepHypRef Expression
1 difss 3124 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssexg 3970 . 2 (((𝐴𝐵) ⊆ 𝐴𝐴𝑉) → (𝐴𝐵) ∈ V)
31, 2mpan 415 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1438  Vcvv 2619  cdif 2994  wss 2997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-in 3003  df-ss 3010
This theorem is referenced by:  frirrg  4168  2oconcl  6185  phplem4dom  6558  fidifsnen  6566  findcard  6584  findcard2  6585  findcard2s  6586  fisseneq  6621  exmidfodomrlemim  6806
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