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Theorem difexg 3939
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 3108 . 2 (𝐴𝐵) ⊆ 𝐴
2 ssexg 3937 . 2 (((𝐴𝐵) ⊆ 𝐴𝐴𝑉) → (𝐴𝐵) ∈ V)
31, 2mpan 415 1 (𝐴𝑉 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  Vcvv 2610  cdif 2979  wss 2982
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995
This theorem is referenced by:  frirrg  4133  2oconcl  6106  phplem4dom  6418  fidifsnen  6426  findcard  6444  findcard2  6445  findcard2s  6446  fisseneq  6474
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