Theorem exists2 1073

Description: A condition implying that at least two things exist.

Assertion

Ref	Expression
exists2	⊢ ((∃xφ ∧ ∃x ¬ φ) → ¬ ∃!x x = x)

Proof of Theorem exists2

Step	Hyp	Ref	Expression
1		exists1 1072	. . 3 ⊢ (∃!x x = x ↔ ∀x x = y)
2		pm3.24 496	. . . 4 ⊢ ¬ (φ ∧ ¬ φ)
3		ax-16 922	. . . . . . 7 ⊢ (∀x x = y → (φ → ∀xφ))
4	3	a5i 687	. . . . . 6 ⊢ (∀x x = y → ∀x(φ → ∀xφ))
5		19.9t 719	. . . . . 6 ⊢ (∀x(φ → ∀xφ) → (∃xφ → φ))
6	4, 5	syl 12	. . . . 5 ⊢ (∀x x = y → (∃xφ → φ))
7		ax-16 922	. . . . . . 7 ⊢ (∀x x = y → (¬ φ → ∀x ¬ φ))
8	7	a5i 687	. . . . . 6 ⊢ (∀x x = y → ∀x(¬ φ → ∀x ¬ φ))
9		19.9t 719	. . . . . 6 ⊢ (∀x(¬ φ → ∀x ¬ φ) → (∃x ¬ φ → ¬ φ))
10	8, 9	syl 12	. . . . 5 ⊢ (∀x x = y → (∃x ¬ φ → ¬ φ))
11	6, 10	anim12d 431	. . . 4 ⊢ (∀x x = y → ((∃xφ ∧ ∃x ¬ φ) → (φ ∧ ¬ φ)))
12	2, 11	mtoi 94	. . 3 ⊢ (∀x x = y → ¬ (∃xφ ∧ ∃x ¬ φ))
13	1, 12	sylbi 174	. 2 ⊢ (∃!x x = x → ¬ (∃xφ ∧ ∃x ¬ φ))
14	13	con2i 89	1 ⊢ ((∃xφ ∧ ∃x ¬ φ) → ¬ ∃!x x = x)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  ∃!weu 1007

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-12 802  ax-16 922

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

metamath.org