Theorem n0moeu 3563

Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.)

Assertion

Ref	Expression
n0moeu	⊢ (A ≠ ∅ → (∃*x x ∈ A ↔ ∃!x x ∈ A))

Distinct variable group:   x,A

Proof of Theorem n0moeu

Step	Hyp	Ref	Expression
1		n0 3560	. . . 4 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
2	1	biimpi 186	. . 3 ⊢ (A ≠ ∅ → ∃x x ∈ A)
3	2	biantrurd 494	. 2 ⊢ (A ≠ ∅ → (∃*x x ∈ A ↔ (∃x x ∈ A ∧ ∃*x x ∈ A)))
4		eu5 2242	. 2 ⊢ (∃!x x ∈ A ↔ (∃x x ∈ A ∧ ∃*x x ∈ A))
5	3, 4	syl6bbr 254	1 ⊢ (A ≠ ∅ → (∃*x x ∈ A ↔ ∃!x x ∈ A))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205   ≠ wne 2517  ∅c0 3551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is used by: (None)