Theorem dfuni2 2509

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 2508	. 2 |- U.A = {x | E.y(x e. y /\ y e. A)}
2		exancom 1056	. . . 4 |- (E.y(x e. y /\ y e. A) <-> E.y(y e. A /\ x e. y))
3		df-rex 1653	. . . 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 1578	. 2 |- {x | E.y(x e. y /\ y e. A)} = {x | E.y e. A x e. y}
6	1, 5	eqtr 1498	1 |- U.A = {x | E.y e. A x e. y}

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  E.wrex 1649  U.cuni 2507

This theorem is referenced by:  uniiun 2605

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rex 1653  df-uni 2508

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