Theorem uni0b 4877

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 4584	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
2	1	ralbii 3084	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
3		dfss3 3911	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
4		neq0 4293	. . . 4 ⊢ (¬ ∪ 𝐴 = ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝐴)
5		rexcom4 3265	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
6		neq0 4293	. . . . . 6 ⊢ (¬ 𝑥 = ∅ ↔ ∃𝑦 𝑦 ∈ 𝑥)
7	6	rexbii 3085	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝑥)
8		eluni2 4855	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
9	8	exbii 1850	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
10	5, 7, 9	3bitr4ri 304	. . . 4 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 = ∅)
11		rexnal 3090	. . . 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 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ cuni 4851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-v 3432  df-dif 3893  df-ss 3907  df-nul 4275  df-sn 4569  df-uni 4852

This theorem is referenced by:  uni0c  4878  uni0OLD  4880  fin1a2lem11  10323  zornn0g  10418  0top  22958  filconn  23858  alexsubALTlem2  24023  ordcmp  36645  unisn0  45503