Theorem uni0b 4957

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 4664	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
2	1	ralbii 3099	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
3		dfss3 3997	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
4		neq0 4375	. . . 4 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝐴)
5		rexcom4 3294	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
6		neq0 4375	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
7	6	rexbii 3100	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥)
8		eluni2 4935	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
9	8	exbii 1846	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
10	5, 7, 9	3bitr4ri 304	. . . 4 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅)
11		rexnal 3106	. . . 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 1777   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-sn 4649  df-uni 4932

This theorem is referenced by:  uni0c  4958  uni0  4959  fin1a2lem11  10479  zornn0g  10574  0top  23011  filconn  23912  alexsubALTlem2  24077  ordcmp  36413  unisn0  44956