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Theorem eqsnuniex 5361
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 4918 . . . . 5 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
2 unieq 4918 . . . . . 6 ({ 𝐴} = ∅ → { 𝐴} = ∅)
3 uni0 4935 . . . . . 6 ∅ = ∅
42, 3eqtrdi 2793 . . . . 5 ({ 𝐴} = ∅ → { 𝐴} = ∅)
51, 4sylan9eq 2797 . . . 4 ((𝐴 = { 𝐴} ∧ { 𝐴} = ∅) → 𝐴 = ∅)
65sneqd 4638 . . 3 ((𝐴 = { 𝐴} ∧ { 𝐴} = ∅) → { 𝐴} = {∅})
7 0inp0 5359 . . . 4 ({ 𝐴} = ∅ → ¬ { 𝐴} = {∅})
87adantl 481 . . 3 ((𝐴 = { 𝐴} ∧ { 𝐴} = ∅) → ¬ { 𝐴} = {∅})
96, 8pm2.65da 817 . 2 (𝐴 = { 𝐴} → ¬ { 𝐴} = ∅)
10 snprc 4717 . . . 4 𝐴 ∈ V ↔ { 𝐴} = ∅)
1110bicomi 224 . . 3 ({ 𝐴} = ∅ ↔ ¬ 𝐴 ∈ V)
1211con2bii 357 . 2 ( 𝐴 ∈ V ↔ ¬ { 𝐴} = ∅)
139, 12sylibr 234 1 (𝐴 = { 𝐴} → 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626   cuni 4907
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-11 2157  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-sn 4627  df-uni 4908
This theorem is referenced by:  en1b  9065  en1uniel  9069
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