Theorem uni0b 3836

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 3443	. . . 4 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
2	1	ralbii 2483	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 = ∅ ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3		ralcom4 2761	. . 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 3147	. . 3 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
6		velsn 3611	. . . 4 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
7	6	ralbii 2483	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
8	5, 7	bitri 184	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
9		eluni2 3815	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
10	9	notbii 668	. . . 4 ⊢ (¬ 𝑦 ∈ ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
11	10	albii 1470	. . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴 ↔ ∀𝑦 ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
12		eq0 3443	. . 3 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ ∪ 𝐴)
13		ralnex 2465	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)
14	13	albii 1470	. . 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 1351   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456   ⊆ wss 3131  ∅c0 3424  {csn 3594  ∪ cuni 3811

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-uni 3812

This theorem is referenced by:  uni0c  3837  uni0  3838