Theorem rmo2 3132

Description: Alternate definition of restricted "at most one." Note that ∃*x ∈ Aφ is not equivalent to ∃y ∈ A∀x ∈ A(φ → x = y) (in analogy to reu6 3026); to see this, let A be the empty set. However, one direction of this pattern holds; see rmo2i 3133. (Contributed by NM, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmo2.1	⊢ Ⅎyφ

Assertion

Ref	Expression
rmo2	⊢ (∃*x ∈ A φ ↔ ∃y∀x ∈ A (φ → x = y))

Distinct variable group:   x,y,A

Allowed substitution hints:   φ(x,y)

Proof of Theorem rmo2

Step	Hyp	Ref	Expression
1		df-rmo 2623	. 2 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
2		nfv 1619	. . . 4 ⊢ Ⅎy x ∈ A
3		rmo2.1	. . . 4 ⊢ Ⅎyφ
4	2, 3	nfan 1824	. . 3 ⊢ Ⅎy(x ∈ A ∧ φ)
5	4	mo2 2233	. 2 ⊢ (∃*x(x ∈ A ∧ φ) ↔ ∃y∀x((x ∈ A ∧ φ) → x = y))
6		impexp 433	. . . . 5 ⊢ (((x ∈ A ∧ φ) → x = y) ↔ (x ∈ A → (φ → x = y)))
7	6	albii 1566	. . . 4 ⊢ (∀x((x ∈ A ∧ φ) → x = y) ↔ ∀x(x ∈ A → (φ → x = y)))
8		df-ral 2620	. . . 4 ⊢ (∀x ∈ A (φ → x = y) ↔ ∀x(x ∈ A → (φ → x = y)))
9	7, 8	bitr4i 243	. . 3 ⊢ (∀x((x ∈ A ∧ φ) → x = y) ↔ ∀x ∈ A (φ → x = y))
10	9	exbii 1582	. 2 ⊢ (∃y∀x((x ∈ A ∧ φ) → x = y) ↔ ∃y∀x ∈ A (φ → x = y))
11	1, 5, 10	3bitri 262	1 ⊢ (∃*x ∈ A φ ↔ ∃y∀x ∈ A (φ → x = y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2615  ∃*wrmo 2618

This theorem was proved from 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

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2620  df-rmo 2623

This theorem is referenced by:  rmo2i  3133