Theorem dfuni2 2505

Description: Alternate definition of class union.

Assertion

Ref	Expression
dfuni2	|- U.A = {x | E.y e. A x e. y}

Distinct variable group:   x,y,A

Proof of Theorem dfuni2

Step	Hyp	Ref	Expression
1		df-uni 2504	. 2 |- U.A = {x | E.y(x e. y /\ y e. A)}
2		exancom 1054	. . . 4 |- (E.y(x e. y /\ y e. A) <-> E.y(y e. A /\ x e. y))
3		df-rex 1650	. . . 4 |- (E.y e. A x e. y <-> E.y(y e. A /\ x e. y))
4	2, 3	bitr4 176	. . 3 |- (E.y(x e. y /\ y e. A) <-> E.y e. A x e. y)
5	4	abbii 1575	. 2 |- {x | E.y(x e. y /\ y e. A)} = {x | E.y e. A x e. y}
6	1, 5	eqtr 1495	1 |- U.A = {x | E.y e. A x e. y}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  U.cuni 2503

This theorem is referenced by:  uniiun 2601

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-rex 1650  df-uni 2504

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