Theorem 2eu5 1496

Description: An alternate definition of double existential uniqueness (see 2eu4 1495). A mistake sometimes made in the literature is to use E!xE!y to mean "exactly one x and exactly one y." (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 A.xE*yph as an additional condition. The correct definition apparently has never been published. (E* means "exists at most one.")

Assertion

Ref	Expression
2eu5	|- ((E!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

Distinct variable groups:   x,y,z,w   ph,z,w

Proof of Theorem 2eu5

Step	Hyp	Ref	Expression
1		2eu1 1492	. . 3 |- (A.xE*yph -> (E!xE!yph <-> (E!xE.yph /\ E!yE.xph)))
2	1	pm5.32ri 657	. 2 |- ((E!xE!yph /\ A.xE*yph) <-> ((E!xE.yph /\ E!yE.xph) /\ A.xE*yph))
3		eumo 1453	. . . . 5 |- (E!yE.xph -> E*yE.xph)
4	3	adantl 397	. . . 4 |- ((E!xE.yph /\ E!yE.xph) -> E*yE.xph)
5		2moex 1483	. . . 4 |- (E*yE.xph -> A.xE*yph)
6	4, 5	syl 10	. . 3 |- ((E!xE.yph /\ E!yE.xph) -> A.xE*yph)
7	6	pm4.71i 648	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> ((E!xE.yph /\ E!yE.xph) /\ A.xE*yph))
8		2eu4 1495	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))
9	2, 7, 8	3bitr2i 186	1 |- ((E!xE!yph /\ A.xE*yph) <-> (E.xE.yph /\ E.zE.wA.xA.y(ph -> (x = z /\ y = w))))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995   = wceq 997  E.wex 1021  E!weu 1422  E*wmo 1423

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425

Copyright terms: Public domain