Theorem uniss2 3632

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 3623	. . . . 5 |- ( ( x C_ y /\ y e. B ) -> x C_ U. B )
2	1	expcom 114	. . . 4 |- ( y e. B -> ( x C_ y -> x C_ U. B ) )
3	2	rexlimiv 2471	. . 3 |- ( E. y e. B x C_ y -> x C_ U. B )
4	3	ralimi 2426	. 2 |- ( A. x e. A E. y e. B x C_ y -> A. x e. A x C_ U. B )
5		unissb 3631	. 2 |- ( U. A C_ U. B <-> A. x e. A x C_ U. B )
6	4, 5	sylibr 132	1 |- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B )

Syntax hints:   -> wi 4   e. wcel 1433   A.wral 2348   E.wrex 2349   C_ wss 2973   U.cuni 3601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602

This theorem is referenced by:  unidif  3633