Theorem ifssun 3590

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 3577	. 2 |- if ( ph , A , B ) = ( { x e. A | ph } u. { x e. B | -. ph } )
2		ssrab2 3282	. . 3 |- { x e. A | ph } C_ A
3		ssrab2 3282	. . 3 |- { x e. B | -. ph } C_ B
4		unss12 3349	. . 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 3229	1 |- if ( ph , A , B ) C_ ( A u. B )

Syntax hints:   -. wn 3   {crab 2489   u. cun 3168   C_ wss 3170   ifcif 3575

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-if 3576

This theorem is referenced by:  ifidss  3591  ifelpwung  4536