Theorem uniss2 3867

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 3858	. . . . 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 2605	. . 3 |- ( E. y e. B x C_ y -> x C_ U. B )
4	3	ralimi 2557	. 2 |- ( A. x e. A E. y e. B x C_ y -> A. x e. A x C_ U. B )
5		unissb 3866	. 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 2164   A.wral 2472   E.wrex 2473   C_ wss 3154   U.cuni 3836

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-in 3160  df-ss 3167  df-uni 3837

This theorem is referenced by:  unidif  3868