Theorem moabex 5444

Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.) Avoid axioms. (Revised by SN, 2-Feb-2026.)

Assertion

Ref	Expression
moabex	⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Proof of Theorem moabex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfmo 2571	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		df-sn 4595	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3		vsnex 5411	. . . . . 6 ⊢ {𝑦} ∈ V
4	2, 3	eqeltrri 2863	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = 𝑦} ∈ V
5	4	a1i 11	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝑥 = 𝑦} ∈ V)
6		ss2abim 4017	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 = 𝑦})
7	5, 6	ssexd 5300	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
8	7	exlimiv 1963	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
9	1, 8	sylbi 220	1 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812   ∈ wcel 2146  ∃*wmo 2568  {cab 2744  Vcvv 3458  {csn 4594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-mo 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-un 3913  df-in 3915  df-ss 3925  df-sn 4595  df-pr 4597

This theorem is used by:  rmorabex  5446  euabex  5447  satfv0  35871