Theorem eqsnuniex 5292

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 4851	. . . . 5 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 = ∪ {∪ 𝐴})
2		unieq 4851	. . . . . 6 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∪ ∅)
3		uni0 4868	. . . . . 6 ⊢ ∪ ∅ = ∅
4	2, 3	eqtrdi 2792	. . . . 5 ⊢ ({∪ 𝐴} = ∅ → ∪ {∪ 𝐴} = ∅)
5	1, 4	sylan9eq 2796	. . . 4 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ∪ 𝐴 = ∅)
6	5	sneqd 4569	. . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → {∪ 𝐴} = {∅})
7		0inp0 5289	. . . 4 ⊢ ({∪ 𝐴} = ∅ → ¬ {∪ 𝐴} = {∅})
8	7	adantl 483	. . 3 ⊢ ((𝐴 = {∪ 𝐴} ∧ {∪ 𝐴} = ∅) → ¬ {∪ 𝐴} = {∅})
9	6, 8	pm2.65da 823	. 2 ⊢ (𝐴 = {∪ 𝐴} → ¬ {∪ 𝐴} = ∅)
10		snprc 4651	. . . 4 ⊢ (¬ ∪ 𝐴 ∈ V ↔ {∪ 𝐴} = ∅)
11	10	bicomi 226	. . 3 ⊢ ({∪ 𝐴} = ∅ ↔ ¬ ∪ 𝐴 ∈ V)
12	11	con2bii 359	. 2 ⊢ (∪ 𝐴 ∈ V ↔ ¬ {∪ 𝐴} = ∅)
13	9, 12	sylibr 236	1 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433  ∅c0 4263  {csn 4557  ∪ cuni 4840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5230

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-v 3435  df-dif 3887  df-ss 3901  df-nul 4264  df-sn 4558  df-uni 4841

This theorem is referenced by:  en1b  8966  en1uniel  8970