Theorem uniss2 3924

Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)

Assertion

Ref	Expression
uniss2	|- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B )

Distinct variable groups:   x, A   x, y, B

Allowed substitution hint:   A( y)

Proof of Theorem uniss2

Step	Hyp	Ref	Expression
1		ssuni 3915	. . . . 5 |- ( ( x C_ y /\ y e. B ) -> x C_ U. B )
2	1	expcom 116	. . . 4 |- ( y e. B -> ( x C_ y -> x C_ U. B ) )
3	2	rexlimiv 2644	. . 3 |- ( E. y e. B x C_ y -> x C_ U. B )
4	3	ralimi 2595	. 2 |- ( A. x e. A E. y e. B x C_ y -> A. x e. A x C_ U. B )
5		unissb 3923	. 2 |- ( U. A C_ U. B <-> A. x e. A x C_ U. B )
6	4, 5	sylibr 134	1 |- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B )

Syntax hints:   -> wi 4   e. wcel 2202   A.wral 2510   E.wrex 2511   C_ wss 3200   U.cuni 3893

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 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

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-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894

This theorem is referenced by:  unidif  3925