Theorem uniss 1936

Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39.

Assertion

Ref	Expression
uniss	|- (A (_ B -> U.A (_ U.B)

Proof of Theorem uniss

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . 6 |- (A (_ B -> (y e. A -> y e. B))
2	1	anim2d 433	. . . . 5 |- (A (_ B -> ((x e. y /\ y e. A) -> (x e. y /\ y e. B)))
3	2	19.22dv 947	. . . 4 |- (A (_ B -> (E.y(x e. y /\ y e. A) -> E.y(x e. y /\ y e. B)))
4	3	19.21aiv 943	. . 3 |- (A (_ B -> A.x(E.y(x e. y /\ y e. A) -> E.y(x e. y /\ y e. B)))
5		ss2ab 1551	. . 3 |- ({x | E.y(x e. y /\ y e. A)} (_ {x | E.y(x e. y /\ y e. B)} <-> A.x(E.y(x e. y /\ y e. A) -> E.y(x e. y /\ y e. B)))
6	4, 5	sylibr 175	. 2 |- (A (_ B -> {x | E.y(x e. y /\ y e. A)} (_ {x | E.y(x e. y /\ y e. B)})
7		df-uni 1920	. 2 |- U.A = {x | E.y(x e. y /\ y e. A)}
8		df-uni 1920	. 2 |- U.B = {x | E.y(x e. y /\ y e. B)}
9	6, 7, 8	3sstr4g 1541	1 |- (A (_ B -> U.A (_ U.B)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803  {cab 1090   e. wcel 1092   (_ wss 1487  U.cuni 1919

This theorem is referenced by:  uni0 1938  unidif 1943  trcl 3489  cflim 3704  dfchsup2 5299  hsupval2t 5301  hsupvalt 5302  shsupclt 5307  hsupss 5310  shsupunss 5316

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-uni 1920

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