Theorem uniss2 3923

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. See iunss2 4012 for a generalization to indexed unions. (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 3914	. . . . 5 |- ((x C_ y /\ y e. B) -> x C_ U.B)
2	1	expcom 424	. . . 4 |- (y e. B -> (x C_ y -> x C_ U.B))
3	2	rexlimiv 2733	. . 3 |- (E.y e. B x C_ y -> x C_ U.B)
4	3	ralimi 2690	. 2 |- (A.x e. A E.y e. B x C_ y -> A.x e. A x C_ U.B)
5		unissb 3922	. 2 |- (U.A C_ U.B <-> A.x e. A x C_ U.B)
6	4, 5	sylibr 203	1 |- (A.x e. A E.y e. B x C_ y -> U.A C_ U.B)

Syntax hints:   -> wi 4   e. wcel 1710  A.wral 2615  E.wrex 2616   C_ wss 3258  U.cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-uni 3893

This theorem is referenced by:  unidif  3924