Theorem unisn2 2870

Description: A version of unisn 2512 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 2513	. . 3 ⊢ (A ∈ V → ∪{A} = A)
2		eqid 1473	. . . . 5 ⊢ A = A
3	2	olci 271	. . . 4 ⊢ (A = ∅ ⋁ A = A)
4		elprg 2419	. . . 4 ⊢ (A ∈ V → (A ∈ {∅, A} ↔ (A = ∅ ⋁ A = A)))
5	3, 4	mpbiri 194	. . 3 ⊢ (A ∈ V → A ∈ {∅, A})
6	1, 5	eqeltrd 1545	. 2 ⊢ (A ∈ V → ∪{A} ∈ {∅, A})
7		snprc 2439	. . . . 5 ⊢ (¬ A ∈ V ↔ {A} = ∅)
8	7	biimp 151	. . . 4 ⊢ (¬ A ∈ V → {A} = ∅)
9	8	unieqd 2507	. . 3 ⊢ (¬ A ∈ V → ∪{A} = ∪∅)
10		uni0 2520	. . . 4 ⊢ ∪∅ = ∅
11		0ex 2706	. . . . 5 ⊢ ∅ ∈ V
12	11	pri1 2446	. . . 4 ⊢ ∅ ∈ {∅, A}
13	10, 12	eqeltr 1541	. . 3 ⊢ ∪∅ ∈ {∅, A}
14	9, 13	syl6eqel 1553	. 2 ⊢ (¬ A ∈ V → ∪{A} ∈ {∅, A})
15	6, 14	pm2.61i 126	1 ⊢ ∪{A} ∈ {∅, A}

Syntax hints:  ¬ wn 2   ⋁ wo 222   = wceq 954   ∈ wcel 956  Vcvv 1807  ∅c0 2276  {csn 2405  {cpr 2406  ∪cuni 2498

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-nul 2705

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-pr 2409  df-uni 2499

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