Theorem eqsnuniex 5367

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 4923	. . . . 5 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 = ∪ {∪ 𝐴})
2		unieq 4923	. . . . . 6 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∪ ∅)
3		uni0 4940	. . . . . 6 ⊢ ∪ ∅ = ∅
4	2, 3	eqtrdi 2791	. . . . 5 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∅)
5	1, 4	sylan9eq 2795	. . . 4 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ∪ 𝐴 = ∅)
6	5	sneqd 4643	. . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → {∪ 𝐴} = {∅})
7		0inp0 5365	. . . 4 ⊢ ({∪ 𝐴} = ∅ → ¬ {∪ 𝐴} = {∅})
8	7	adantl 481	. . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ¬ {∪ 𝐴} = {∅})
9	6, 8	pm2.65da 817	. 2 ⊢ (𝐴 = {∪ 𝐴} → ¬ {∪ 𝐴} = ∅)
10		snprc 4722	. . . 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 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  {csn 4631  ∪ cuni 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-sn 4632  df-uni 4913

This theorem is referenced by:  en1b  9064  en1uniel  9068