Theorem uniss2 4470

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 4565 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 4459	. . . . 5 |- ( ( x C_ y /\ y e. B ) -> x C_ U. B )
2	1	expcom 451	. . . 4 |- ( y e. B -> ( x C_ y -> x C_ U. B ) )
3	2	rexlimiv 3027	. . 3 |- ( E. y e. B x C_ y -> x C_ U. B )
4	3	ralimi 2952	. 2 |- ( A. x e. A E. y e. B x C_ y -> A. x e. A x C_ U. B )
5		unissb 4469	. 2 |- ( U. A C_ U. B <-> A. x e. A x C_ U. B )
6	4, 5	sylibr 224	1 |- ( A. x e. A E. y e. B x C_ y -> U. A C_ U. B )

Syntax hints:   -> wi 4   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-uni 4437

This theorem is referenced by:  unidif  4471  coflim  9083