Theorem unisn2 4794

Description: A version of unisn 4451 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 4452	. . 3 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
2		prid2g 4296	. . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
3	1, 2	eqeltrd 2701	. 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
4		snprc 4253	. . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5	4	biimpi 206	. . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
6	5	unieqd 4446	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅)
7		uni0 4465	. . . 4 ⊢ ∪ ∅ = ∅
8		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
9	8	prid1 4297	. . . 4 ⊢ ∅ ∈ {∅, 𝐴}
10	7, 9	eqeltri 2697	. . 3 ⊢ ∪ ∅ ∈ {∅, 𝐴}
11	6, 10	syl6eqel 2709	. 2 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} ∈ {∅, 𝐴})
12	3, 11	pm2.61i 176	1 ⊢ ∪ {𝐴} ∈ {∅, 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177  {cpr 4179  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by: (None)