Theorem eqsnuniex 5319

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

Step	Hyp	Ref	Expression
1		unieq 4885	. . . . 5 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 = ∪ {∪ 𝐴})
2		unieq 4885	. . . . . 6 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∪ ∅)
3		uni0 4902	. . . . . 6 ⊢ ∪ ∅ = ∅
4	2, 3	eqtrdi 2781	. . . . 5 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∅)
5	1, 4	sylan9eq 2785	. . . 4 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ∪ 𝐴 = ∅)
6	5	sneqd 4604	. . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → {∪ 𝐴} = {∅})
7		0inp0 5317	. . . 4 ⊢ ({∪ 𝐴} = ∅ → ¬ {∪ 𝐴} = {∅})
8	7	adantl 481	. . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ¬ {∪ 𝐴} = {∅})
9	6, 8	pm2.65da 816	. 2 ⊢ (𝐴 = {∪ 𝐴} → ¬ {∪ 𝐴} = ∅)
10		snprc 4684	. . . 4 ⊢ (¬ ∪ 𝐴 ∈ V ↔ {∪ 𝐴} = ∅)
11	10	bicomi 224	. . 3 ⊢ ({∪ 𝐴} = ∅ ↔ ¬ ∪ 𝐴 ∈ V)
12	11	con2bii 357	. 2 ⊢ (∪ 𝐴 ∈ V ↔ ¬ {∪ 𝐴} = ∅)
13	9, 12	sylibr 234	1 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299  {csn 4592  ∪ cuni 4874

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-v 3452  df-dif 3920  df-ss 3934  df-nul 4300  df-sn 4593  df-uni 4875

This theorem is referenced by:  en1b  8999  en1uniel  9003