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Theorem eqsnuniex 5352
Description: If a class is equal to the singleton of its union, then its union exists. (Contributed by BTernaryTau, 24-Sep-2024.)
Assertion
Ref Expression
eqsnuniex (𝐴 = { 𝐴} → 𝐴 ∈ V)

Proof of Theorem eqsnuniex
StepHypRef Expression
1 unieq 4913 . . . . 5 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
2 unieq 4913 . . . . . 6 ({ 𝐴} = ∅ → { 𝐴} = ∅)
3 uni0 4932 . . . . . 6 ∅ = ∅
42, 3eqtrdi 2782 . . . . 5 ({ 𝐴} = ∅ → { 𝐴} = ∅)
51, 4sylan9eq 2786 . . . 4 ((𝐴 = { 𝐴} ∧ { 𝐴} = ∅) → 𝐴 = ∅)
65sneqd 4635 . . 3 ((𝐴 = { 𝐴} ∧ { 𝐴} = ∅) → { 𝐴} = {∅})
7 0inp0 5350 . . . 4 ({ 𝐴} = ∅ → ¬ { 𝐴} = {∅})
87adantl 481 . . 3 ((𝐴 = { 𝐴} ∧ { 𝐴} = ∅) → ¬ { 𝐴} = {∅})
96, 8pm2.65da 814 . 2 (𝐴 = { 𝐴} → ¬ { 𝐴} = ∅)
10 snprc 4716 . . . 4 𝐴 ∈ V ↔ { 𝐴} = ∅)
1110bicomi 223 . . 3 ({ 𝐴} = ∅ ↔ ¬ 𝐴 ∈ V)
1211con2bii 357 . 2 ( 𝐴 ∈ V ↔ ¬ { 𝐴} = ∅)
139, 12sylibr 233 1 (𝐴 = { 𝐴} → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  {csn 4623   cuni 4902
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-11 2146  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-v 3470  df-dif 3946  df-in 3950  df-ss 3960  df-nul 4318  df-sn 4624  df-uni 4903
This theorem is referenced by:  en1b  9025  en1uniel  9030
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