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Theorem moabex 5083
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 2565 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abss 3831 . . . . 5 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 4350 . . . . . . 7 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43imbi2i 327 . . . . . 6 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1914 . . . . 5 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 266 . . . 4 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
7 snex 5064 . . . . 5 {𝑦} ∈ V
87ssex 4963 . . . 4 ({𝑥𝜑} ⊆ {𝑦} → {𝑥𝜑} ∈ V)
96, 8sylbir 226 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
109exlimiv 2025 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
111, 10sylbi 208 1 (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650  wex 1874  wcel 2155  ∃*wmo 2563  {cab 2751  Vcvv 3350  wss 3732  {csn 4334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-sn 4335  df-pr 4337
This theorem is referenced by:  rmorabex  5084  euabex  5085
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