Theorem 2eu5 1071

Description: An alternate definition of double existential uniqueness (see 2eu4 1070). 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 group(s):   x,y,z,w   φ,z,w

Proof of Theorem 2eu5

Step	Hyp	Ref	Expression
1		eumo 1037	. . . . . 6 ⊢ (∃!y∃xφ → ∃*y∃xφ)
2	1	adantl 305	. . . . 5 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∃*y∃xφ)
3		2moex 1060	. . . . 5 ⊢ (∃*y∃xφ → ∀x∃*yφ)
4	2, 3	syl 12	. . . 4 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) → ∀x∃*yφ)
5	4	pm4.71i 483	. . 3 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ ((∃!x∃yφ ∧ ∃!y∃xφ) ∧ ∀x∃*yφ))
6		2eu1 1067	. . . 4 ⊢ (∀x∃*yφ → (∃!x∃!yφ ↔ (∃!x∃yφ ∧ ∃!y∃xφ)))
7	6	pm5.32ri 490	. . 3 ⊢ ((∃!x∃!yφ ∧ ∀x∃*yφ) ↔ ((∃!x∃yφ ∧ ∃!y∃xφ) ∧ ∀x∃*yφ))
8	5, 7	bitr4 154	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ (∃!x∃!yφ ∧ ∀x∃*yφ))
9		2eu4 1070	. 2 ⊢ ((∃!x∃yφ ∧ ∃!y∃xφ) ↔ (∃x∃yφ ∧ ∃z∃w∀x∀y(φ → (x = z ∧ y = w))))
10	8, 9	bitr3 153	1 ⊢ ((∃!x∃!yφ ∧ ∀x∃*yφ) ↔ (∃x∃yφ ∧ ∃z∃w∀x∀y(φ → (x = z ∧ y = w))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007  ∃*wmo 1008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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