Theorem 2eu5 2382

Description: An alternate definition of double existential uniqueness (see 2eu4 2380). A mistake sometimes made in the literature is to use E!xE! to mean "exactly one and exactly one ." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining as an additional condition. The correct definition apparently has never been published. (E* means "exists at most one.") (Contributed by NM, 26-Oct-2003.)

Assertion

Ref	Expression
2eu5

Distinct variable groups:   ,,,   ,,

Proof of Theorem 2eu5

Step	Hyp	Ref	Expression
1		2eu1 2376	. . 3
2	1	pm5.32ri 638	. 2
3		eumo 2313	. . . . 5
4	3	adantl 466	. . . 4
5		2moex 2365	. . . 4
6	4, 5	syl 16	. . 3
7	6	pm4.71i 632	. 2
8		2eu4 2380	. 2
9	2, 7, 8	3bitr2i 273	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  E!weu 2282  E*wmo 2283

This theorem is referenced by:  2reu5lem3  3307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287