Theorem uni0b 3874

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 3478	. . . 4 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
2	1	ralbii 2511	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3		ralcom4 2793	. . 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 3181	. . 3 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
6		velsn 3649	. . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
7	6	ralbii 2511	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
8	5, 7	bitri 184	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
9		eluni2 3853	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
10	9	notbii 669	. . . 4 ⊢ (¬ 𝑦 ∈ ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
11	10	albii 1492	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴 ↔ ∀𝑦 ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
12		eq0 3478	. . 3 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴)
13		ralnex 2493	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
14	13	albii 1492	. . 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 1370   = wceq 1372   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484   ⊆ wss 3165  ∅c0 3459  {csn 3632  ∪ cuni 3849

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-uni 3850

This theorem is referenced by:  uni0c  3875  uni0  3876