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Theorem difexg 3923
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 3095 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssexg 3921 . 2 (((𝐴𝐵) ⊆ 𝐴𝐴𝑉) → (𝐴𝐵) ∈ V)
31, 2mpan 408 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1407  Vcvv 2572  cdif 2939  wss 2942
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 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-v 2574  df-dif 2945  df-in 2949  df-ss 2956
This theorem is referenced by:  frirrg  4112  2oconcl  6050  phplem4dom  6352  fidifsnen  6359  findcard  6373  findcard2  6374  findcard2s  6375
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