Theorem ifssun 3563

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 3551	. 2 |- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )
2		ssrab2 3255	. . 3 |- { x e. A | ph } C_ A
3		ssrab2 3255	. . 3 |- { x e. B | -. ph } C_ B
4		unss12 3322	. . 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 3202	1 |- if ( ph , A , B ) C_ ( A u. B )

Syntax hints:   -. wn 3   {crab 2472   u. cun 3142   C_ wss 3144   ifcif 3549

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-if 3550

This theorem is referenced by:  ifidss  3564  ifelpwung  4499