Theorem difexi 4842

Description: Existence of a difference, inference version of difexg 4841. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)

Hypothesis

Ref	Expression
difexi.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
difexi	⊢ (𝐴 ∖ 𝐵) ∈ V

Proof of Theorem difexi

Step	Hyp	Ref	Expression
1		difexi.1	. 2 ⊢ 𝐴 ∈ V
2		difexg 4841	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∖ 𝐵) ∈ V)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∖ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621

This theorem is referenced by:  marypha1lem  8380  inf3lem3  8565  kmlem11  9020  kmlem12  9021  opfi1uzind  13321  uhgrspan1lem1  26237  upgrres1lem1  26246  nbgrval  26274  nbfusgrlevtxm1  26323  vtxdginducedm1lem1  26491  vtxdginducedm1fi  26496  finsumvtxdg2ssteplem4  26500  setindtr  37908  ssdifcl  38193  clsk3nimkb  38655  meaiuninclem  41015  meaiininclem  41021