Theorem unisn2 5218

Description: A version of unisn 4860 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 4859	. . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
2		prid2g 4699	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
3	1, 2	eqeltrd 2915	. 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
4		snprc 4655	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5	4	biimpi 218	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
6	5	unieqd 4854	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅)
7		uni0 4868	. . . 4 ⊢ ∪ ∅ = ∅
8		0ex 5213	. . . . 5 ⊢ ∅ ∈ V
9	8	prid1 4700	. . . 4 ⊢ ∅ ∈ {∅, 𝐴}
10	7, 9	eqeltri 2911	. . 3 ⊢ ∪ ∅ ∈ {∅, 𝐴}
11	6, 10	eqeltrdi 2923	. 2 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
12	3, 11	pm2.61i 184	1 ⊢ ∪ {𝐴} ∈ {∅, 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ∅c0 4293  {csn 4569  {cpr 4571  ∪ cuni 4840

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-sn 4570  df-pr 4572  df-uni 4841

This theorem is referenced by: (None)