Theorem uni0b 3939

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		eq0 3527	. . . 4 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
2	1	ralbii 2548	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3		ralcom4 2836	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥)
4	2, 3	bitri 184	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∀𝑦∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥)
5		dfss3 3227	. . 3 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
6		velsn 3706	. . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
7	6	ralbii 2548	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
8	5, 7	bitri 184	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
9		eluni2 3918	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
10	9	notbii 674	. . . 4 ⊢ (¬ 𝑦 ∈ ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
11	10	albii 1519	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴 ↔ ∀𝑦 ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
12		eq0 3527	. . 3 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴)
13		ralnex 2530	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
14	13	albii 1519	. . 3 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦 ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
15	11, 12, 14	3bitr4i 212	. 2 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑦∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥)
16	4, 8, 15	3bitr4ri 213	1 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1396   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521   ⊆ wss 3211  ∅c0 3508  {csn 3689  ∪ cuni 3914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-uni 3915

This theorem is referenced by:  uni0c  3940  uni0  3941