Theorem moabex 5431

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 2568	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		df-sn 4589	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3		vsnex 5398	. . . . . 6 ⊢ {𝑦} ∈ V
4	2, 3	eqeltrri 2861	. . . . 5 ⊢ {𝑥 ∣ 𝑥 = 𝑦} ∈ V
5	4	a1i 11	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝑥 = 𝑦} ∈ V)
6		ss2abim 3988	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ⊆ {𝑥 ∣ 𝑥 = 𝑦})
7	5, 6	ssexd 5291	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} ∈ V)
8	7	exlimiv 1965	. 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 2147  ∃*wmo 2565  {cab 2741  Vcvv 3454  {csn 4588

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-un 4154  df-sn 4589  df-pr 4591

This theorem is used by:  rmorabex  5433  euabex  5434  satfv0  36303