Theorem uni0b 4938

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 4647	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
2	1	ralbii 3091	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
3		dfss3 3984	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
4		neq0 4358	. . . 4 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝐴)
5		rexcom4 3286	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
6		neq0 4358	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
7	6	rexbii 3092	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥)
8		eluni2 4916	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
9	8	exbii 1845	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
10	5, 7, 9	3bitr4ri 304	. . . 4 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅)
11		rexnal 3098	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
12	4, 10, 11	3bitri 297	. . 3 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
13	12	con4bii 321	. 2 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
14	2, 3, 13	3bitr4ri 304	1 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537  ∃wex 1776   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  ∅c0 4339  {csn 4631  ∪ cuni 4912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-sn 4632  df-uni 4913

This theorem is referenced by:  uni0c  4939  uni0  4940  fin1a2lem11  10448  zornn0g  10543  0top  23006  filconn  23907  alexsubALTlem2  24072  ordcmp  36430  unisn0  44994