Theorem uniss2 3775

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 3766	. . . . 5 |- ( ( x C_ y /\ y e. B ) -> x C_ U. B )
2	1	expcom 115	. . . 4 |- ( y e. B -> ( x C_ y -> x C_ U. B ) )
3	2	rexlimiv 2546	. . 3 |- ( E. y e. B x C_ y -> x C_ U. B )
4	3	ralimi 2498	. 2 |- ( A. x e. A E. y e. B x C_ y -> A. x e. A x C_ U. B )
5		unissb 3774	. 2 |- ( U. A C_ U. B <-> A. x e. A x C_ U. B )
6	4, 5	sylibr 133	1 |- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B )

Syntax hints:   -> wi 4   e. wcel 1481   A.wral 2417   E.wrex 2418   C_ wss 3076   U.cuni 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089  df-uni 3745

This theorem is referenced by:  unidif  3776