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Theorem moabex 5450
Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
Assertion
Ref Expression
moabex (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)

Proof of Theorem moabex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-mo 2526 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abss 4050 . . . . 5 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 4637 . . . . . . 7 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43imbi2i 336 . . . . . 6 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1813 . . . . 5 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 275 . . . 4 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
7 vsnex 5420 . . . . 5 {𝑦} ∈ V
87ssex 5312 . . . 4 ({𝑥𝜑} ⊆ {𝑦} → {𝑥𝜑} ∈ V)
96, 8sylbir 234 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
109exlimiv 1925 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
111, 10sylbi 216 1 (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wex 1773  wcel 2098  ∃*wmo 2524  {cab 2701  Vcvv 3466  wss 3941  {csn 4621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rab 3425  df-v 3468  df-un 3946  df-in 3948  df-ss 3958  df-sn 4622  df-pr 4624
This theorem is referenced by:  rmorabex  5451  euabex  5452  satfv0  34840
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