Theorem unisn2 5252

Description: A version of unisn 4877 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)

Assertion

Ref	Expression
unisn2	⊢ ∪ {𝐴} ∈ {∅, 𝐴}

Proof of Theorem unisn2

Step	Hyp	Ref	Expression
1		unisng 4876	. . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
2		prid2g 4713	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
3	1, 2	eqeltrd 2833	. 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
4		snprc 4669	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5	4	biimpi 216	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
6	5	unieqd 4871	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅)
7		uni0 4886	. . . 4 ⊢ ∪ ∅ = ∅
8		0ex 5247	. . . . 5 ⊢ ∅ ∈ V
9	8	prid1 4714	. . . 4 ⊢ ∅ ∈ {∅, 𝐴}
10	7, 9	eqeltri 2829	. . 3 ⊢ ∪ ∅ ∈ {∅, 𝐴}
11	6, 10	eqeltrdi 2841	. 2 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
12	3, 11	pm2.61i 182	1 ⊢ ∪ {𝐴} ∈ {∅, 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  {csn 4575  {cpr 4577  ∪ cuni 4858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5246

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-sn 4576  df-pr 4578  df-uni 4859

This theorem is referenced by: (None)