Theorem difexg 4231

Description: Existence of a difference. (Contributed by NM, 26-May-1998.)

Assertion

Ref	Expression
difexg	|- ( A e. V -> ( A \ B ) e. _V )

Proof of Theorem difexg

Step	Hyp	Ref	Expression
1		difss 3333	. 2 |- ( A \ B ) C_ A
2		ssexg 4228	. 2 |- ( ( ( A \ B ) C_ A /\ A e. V ) -> ( A \ B ) e. _V )
3	1, 2	mpan 424	1 |- ( A e. V -> ( A \ B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2202   _Vcvv 2802   \ cdif 3197   C_ wss 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-sep 4207

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213

This theorem is referenced by:  frirrg  4447  2oconcl  6606  phplem4dom  7047  fidifsnen  7056  findcard  7076  findcard2  7077  findcard2s  7078  fisseneq  7126  difinfsn  7298  ismkvnex  7353  exmidfodomrlemim  7411