Theorem uniss2 2525

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 2591 for a generalization to indexed unions.

Assertion

Ref	Expression
uniss2	|- (A.x e. A E.y e. B x (_ y -> U.A (_ U.B)

Distinct variable groups:   x,A   x,y,B

Proof of Theorem uniss2

Step	Hyp	Ref	Expression
1		ssuni 2518	. . . . 5 |- ((x (_ y /\ y e. B) -> x (_ U.B)
2	1	expcom 374	. . . 4 |- (y e. B -> (x (_ y -> x (_ U.B))
3	2	r19.23aiv 1741	. . 3 |- (E.y e. B x (_ y -> x (_ U.B)
4	3	r19.20si 1704	. 2 |- (A.x e. A E.y e. B x (_ y -> A.x e. A x (_ U.B)
5		unissb 2524	. 2 |- (U.A (_ U.B <-> A.x e. A x (_ U.B)
6	4, 5	sylibr 200	1 |- (A.x e. A E.y e. B x (_ y -> U.A (_ U.B)

Syntax hints:   -> wi 3   e. wcel 957  A.wral 1643  E.wrex 1644   (_ wss 2044  U.cuni 2499

This theorem is referenced by:  unidif 2526

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-rex 1648  df-v 1809  df-in 2048  df-ss 2050  df-uni 2500

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