Theorem unisn2 4722

Description: A version of unisn 4387 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 4388	. . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
2		prid2g 4240	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
3	1, 2	eqeltrd 2688	. 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
4		snprc 4197	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5	4	biimpi 205	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
6	5	unieqd 4382	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅)
7		uni0 4401	. . . 4 ⊢ ∪ ∅ = ∅
8		0ex 4718	. . . . 5 ⊢ ∅ ∈ V
9	8	prid1 4241	. . . 4 ⊢ ∅ ∈ {∅, 𝐴}
10	7, 9	eqeltri 2684	. . 3 ⊢ ∪ ∅ ∈ {∅, 𝐴}
11	6, 10	syl6eqel 2696	. 2 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
12	3, 11	pm2.61i 175	1 ⊢ ∪ {𝐴} ∈ {∅, 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373

This theorem is referenced by: (None)