Theorem difexg 4236

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 3335	. 2 |- ( A \ B ) C_ A
2		ssexg 4233	. 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 2803   \ cdif 3198   C_ wss 3201

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214

This theorem is referenced by:  frirrg  4453  2oconcl  6650  phplem4dom  7091  fidifsnen  7100  findcard  7120  findcard2  7121  findcard2s  7122  fisseneq  7170  difinfsn  7359  ismkvnex  7414  exmidfodomrlemim  7472