Theorem uni0b 4879

Description: The union of a set is empty iff the set is included in the singleton of the empty set. (Contributed by NM, 12-Sep-2004.)

Assertion

Ref	Expression
uni0b	⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})

Proof of Theorem uni0b

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 4587	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
2	1	ralbii 3093	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
3		dfss3 3919	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
4		neq0 4290	. . . 4 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝐴)
5		rexcom4 3268	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
6		neq0 4290	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
7	6	rexbii 3094	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥)
8		eluni2 4854	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
9	8	exbii 1849	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
10	5, 7, 9	3bitr4ri 303	. . . 4 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅)
11		rexnal 3100	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
12	4, 10, 11	3bitri 296	. . 3 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
13	12	con4bii 320	. 2 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
14	2, 3, 13	3bitr4ri 303	1 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∀wral 3062  ∃wrex 3071   ⊆ wss 3897  ∅c0 4267  {csn 4571  ∪ cuni 4850

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-11 2153  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-v 3443  df-dif 3900  df-in 3904  df-ss 3914  df-nul 4268  df-sn 4572  df-uni 4851

This theorem is referenced by:  uni0c  4880  uni0  4881  fin1a2lem11  10239  zornn0g  10334  0top  22205  filconn  23106  alexsubALTlem2  23271  ordcmp  34694  unisn0  42823