Theorem 2eu5 1451

Description: An alternate definition of double existential uniqueness (see 2eu4 1450). A mistake sometimes made in the literature is to use ∃!x∃!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 ∀x∃*yφ as an additional condition. The correct definition apparently has never been published. (∃* means "exists at most one.")

Assertion

Ref	Expression
2eu5	⊢ ((∃!x∃!yφ ⋀ ∀x∃*yφ) ↔ (∃x∃yφ ⋀ ∃z∃w∀x∀y(φ → (x = z ⋀ y = w))))

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

Proof of Theorem 2eu5

Step	Hyp	Ref	Expression
1		2eu1 1447	. . 3 ⊢ (∀x∃*yφ → (∃!x∃!yφ ↔ (∃!x∃yφ ⋀ ∃!y∃xφ)))
2	1	pm5.32ri 645	. 2 ⊢ ((∃!x∃!yφ ⋀ ∀x∃*yφ) ↔ ((∃!x∃yφ ⋀ ∃!y∃xφ) ⋀ ∀x∃*yφ))
3		eumo 1409	. . . . 5 ⊢ (∃!y∃xφ → ∃*y∃xφ)
4	3	adantl 388	. . . 4 ⊢ ((∃!x∃yφ ⋀ ∃!y∃xφ) → ∃*y∃xφ)
5		2moex 1438	. . . 4 ⊢ (∃*y∃xφ → ∀x∃*yφ)
6	4, 5	syl 10	. . 3 ⊢ ((∃!x∃yφ ⋀ ∃!y∃xφ) → ∀x∃*yφ)
7	6	pm4.71i 636	. 2 ⊢ ((∃!x∃yφ ⋀ ∃!y∃xφ) ↔ ((∃!x∃yφ ⋀ ∃!y∃xφ) ⋀ ∀x∃*yφ))
8		2eu4 1450	. 2 ⊢ ((∃!x∃yφ ⋀ ∃!y∃xφ) ↔ (∃x∃yφ ⋀ ∃z∃w∀x∀y(φ → (x = z ⋀ y = w))))
9	2, 7, 8	3bitr2 179	1 ⊢ ((∃!x∃!yφ ⋀ ∀x∃*yφ) ↔ (∃x∃yφ ⋀ ∃z∃w∀x∀y(φ → (x = z ⋀ y = w))))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 952   = wceq 954  ∃wex 978  ∃!weu 1378  ∃*wmo 1379

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381

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