Theorem unisngOLD 4675

Description: Obsolete proof of unisng 4672 as of 16-Sep-2022. (Contributed by NM, 13-Aug-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
unisngOLD	⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Proof of Theorem unisngOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4406	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	unieqd 4667	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥} = ∪ {𝐴})
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2839	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥} = 𝑥 ↔ ∪ {𝐴} = 𝐴))
5		vex 3416	. . 3 ⊢ 𝑥 ∈ V
6	5	unisnOLD 4674	. 2 ⊢ ∪ {𝑥} = 𝑥
7	4, 6	vtoclg 3481	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  {csn 4396  ∪ cuni 4657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2802

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-rex 3122  df-v 3415  df-un 3802  df-sn 4397  df-pr 4399  df-uni 4658

This theorem is referenced by: (None)