Theorem unisn2 5273

Description: A version of unisn 4889 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 4888	. . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
2		prid2g 4725	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
3	1, 2	eqeltrd 2862	. 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
4		snprc 4681	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5	4	biimpi 219	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
6	5	unieqd 4883	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅)
7		uni0 4899	. . . 4 ⊢ ∪ ∅ = ∅
8		0ex 5268	. . . . 5 ⊢ ∅ ∈ V
9	8	prid1 4726	. . . 4 ⊢ ∅ ∈ {∅, 𝐴}
10	7, 9	eqeltri 2858	. . 3 ⊢ ∪ ∅ ∈ {∅, 𝐴}
11	6, 10	eqeltrdi 2870	. 2 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
12	3, 11	pm2.61i 184	1 ⊢ ∪ {𝐴} ∈ {∅, 𝐴}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ∅c0 4282  {csn 4587  {cpr 4589  ∪ cuni 4870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-sn 4588  df-pr 4590  df-uni 4871

This theorem is used by: (None)