Theorem difexg 1703

Description: Existence of a difference.

Assertion

Ref	Expression
difexg	⊢ (A ∈ C → (A ∖ B) ∈ V)

Proof of Theorem difexg

Step	Hyp	Ref	Expression
1		difss 1596	. 2 ⊢ (A ∖ B) ⊆ A
2		ssexg 1702	. 2 ⊢ (A ∈ C → ((A ∖ B) ⊆ A → (A ∖ B) ∈ V))
3	1, 2	mpi 44	1 ⊢ (A ∈ C → (A ∖ B) ∈ V)

Syntax hints:   → wi 2   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ⊆ wss 1487

This theorem is referenced by:  difex2 1951  oev 3122  limensuci 3401  unfilem3 3440  inf5 3472  kmlem10 3589  kmlem11 3590  fodomb 3615  infxpidmlem12 4944  infdif 4948

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492

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