Theorem ifssun 3550

Description: A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)

Assertion

Ref	Expression
ifssun	|- if ( ph , A , B ) C_ ( A u. B )

Proof of Theorem ifssun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfif6 3538	. 2 |- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )
2		ssrab2 3242	. . 3 |- { x e. A | ph } C_ A
3		ssrab2 3242	. . 3 |- { x e. B | -. ph } C_ B
4		unss12 3309	. . 3 |- ( ( { x e. A | ph } C_ A /\ { x e. B | -. ph } C_ B ) -> ( { x e. A | ph } u. { x e. B | -. ph } ) C_ ( A u. B ) )
5	2, 3, 4	mp2an 426	. 2 |- ( { x e. A | ph } u. { x e. B | -. ph } ) C_ ( A u. B )
6	1, 5	eqsstri 3189	1 |- if ( ph , A , B ) C_ ( A u. B )

Syntax hints:   -. wn 3   {crab 2459   u. cun 3129   C_ wss 3131   ifcif 3536

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-if 3537

This theorem is referenced by:  ifidss  3551  ifelpwung  4483