Theorem uniss2 2543

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 2609 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 2536	. . . . 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 1750	. . 3 |- (E.y e. B x (_ y -> x (_ U.B)
4	3	r19.20si 1713	. 2 |- (A.x e. A E.y e. B x (_ y -> A.x e. A x (_ U.B)
5		unissb 2542	. 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 962  A.wral 1652  E.wrex 1653   (_ wss 2058  U.cuni 2517

This theorem is referenced by:  unidif 2544

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-ral 1656  df-rex 1657  df-v 1819  df-in 2062  df-ss 2064  df-uni 2518

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