MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  moabex Structured version   Visualization version   GIF version

Theorem moabex 5464
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 2540 . 2 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
2 abss 4063 . . . . 5 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 ∈ {𝑦}))
3 velsn 4642 . . . . . . 7 (𝑥 ∈ {𝑦} ↔ 𝑥 = 𝑦)
43imbi2i 336 . . . . . 6 ((𝜑𝑥 ∈ {𝑦}) ↔ (𝜑𝑥 = 𝑦))
54albii 1819 . . . . 5 (∀𝑥(𝜑𝑥 ∈ {𝑦}) ↔ ∀𝑥(𝜑𝑥 = 𝑦))
62, 5bitri 275 . . . 4 ({𝑥𝜑} ⊆ {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
7 vsnex 5434 . . . . 5 {𝑦} ∈ V
87ssex 5321 . . . 4 ({𝑥𝜑} ⊆ {𝑦} → {𝑥𝜑} ∈ V)
96, 8sylbir 235 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
109exlimiv 1930 . 2 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → {𝑥𝜑} ∈ V)
111, 10sylbi 217 1 (∃*𝑥𝜑 → {𝑥𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  wex 1779  wcel 2108  ∃*wmo 2538  {cab 2714  Vcvv 3480  wss 3951  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-un 3956  df-in 3958  df-ss 3968  df-sn 4627  df-pr 4629
This theorem is referenced by:  rmorabex  5465  euabex  5466  satfv0  35363
  Copyright terms: Public domain W3C validator