Theorem uniex2 4579

Description: The Axiom of Union using the standard abbreviation for union. Given any set x, its union y exists. (Contributed by NM, 4-Jun-2006.) (Proof shortened by BJ, 14-Jul-2026.)

Assertion

Ref	Expression
uniex2	|- E. y y = U. x

Distinct variable group:   x, y

Proof of Theorem uniex2

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axun2 4578	. . 3 |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )
2		eluni 3936	. . . . . 6 |- ( z e. U. x <-> E. w ( z e. w /\ w e. x ) )
3	2	bibi2i 227	. . . . 5 |- ( ( z e. y <-> z e. U. x ) <-> ( z e. y <-> E. w ( z e. w /\ w e. x ) ) )
4	3	albii 1523	. . . 4 |- ( A. z ( z e. y <-> z e. U. x ) <-> A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) ) )
5	4	exbii 1658	. . 3 |- ( E. y A. z ( z e. y <-> z e. U. x ) <-> E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) ) )
6	1, 5	mpbir 146	. 2 |- E. y A. z ( z e. y <-> z e. U. x )
7		dfcleq 2232	. . 3 |- ( y = U. x <-> A. z ( z e. y <-> z e. U. x ) )
8	7	biimpri 133	. 2 |- ( A. z ( z e. y <-> z e. U. x ) -> y = U. x )
9	6, 8	eximii 1655	1 |- E. y y = U. x

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1400   = wceq 1402   E.wex 1545   e. wcel 2209   U.cuni 3933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-uni 3934

This theorem is referenced by:  uniex  4581