Theorem moabex 5424

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 2566	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		df-sn 4582	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3		vsnex 5391	. . . . . 6 ⊢ {𝑦} ∈ V
4	2, 3	eqeltrri 2858	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = 𝑦} ∈ V
5	4	a1i 11	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝑥 = 𝑦} ∈ V)
6		ss2abim 4013	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 = 𝑦})
7	5, 6	ssexd 5279	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
8	7	exlimiv 1949	. 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
9	1, 8	sylbi 219	1 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V)

Syntax hints:   → wi 4  ∀wal 1557  ∃wex 1798   ∈ wcel 2141  ∃*wmo 2563  {cab 2739  Vcvv 3453  {csn 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-un 3909  df-in 3911  df-ss 3921  df-sn 4582  df-pr 4584

This theorem is referenced by:  rmorabex  5426  euabex  5427  satfv0  35672