Theorem uni0b 3916

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		elsn 3748	. . 3 ⊢ (x ∈ {∅} ↔ x = ∅)
2	1	ralbii 2638	. 2 ⊢ (∀x ∈ A x ∈ {∅} ↔ ∀x ∈ A x = ∅)
3		dfss3 3263	. 2 ⊢ (A ⊆ {∅} ↔ ∀x ∈ A x ∈ {∅})
4		neq0 3560	. . . 4 ⊢ (¬ ∪A = ∅ ↔ ∃y y ∈ ∪A)
5		rexcom4 2878	. . . . 5 ⊢ (∃x ∈ A ∃y y ∈ x ↔ ∃y∃x ∈ A y ∈ x)
6		neq0 3560	. . . . . 6 ⊢ (¬ x = ∅ ↔ ∃y y ∈ x)
7	6	rexbii 2639	. . . . 5 ⊢ (∃x ∈ A ¬ x = ∅ ↔ ∃x ∈ A ∃y y ∈ x)
8		eluni2 3895	. . . . . 6 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
9	8	exbii 1582	. . . . 5 ⊢ (∃y y ∈ ∪A ↔ ∃y∃x ∈ A y ∈ x)
10	5, 7, 9	3bitr4ri 269	. . . 4 ⊢ (∃y y ∈ ∪A ↔ ∃x ∈ A ¬ x = ∅)
11		rexnal 2625	. . . 4 ⊢ (∃x ∈ A ¬ x = ∅ ↔ ¬ ∀x ∈ A x = ∅)
12	4, 10, 11	3bitri 262	. . 3 ⊢ (¬ ∪A = ∅ ↔ ¬ ∀x ∈ A x = ∅)
13	12	con4bii 288	. 2 ⊢ (∪A = ∅ ↔ ∀x ∈ A x = ∅)
14	2, 3, 13	3bitr4ri 269	1 ⊢ (∪A = ∅ ↔ A ⊆ {∅})

Syntax hints:  ¬ wn 3   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   ⊆ wss 3257  ∅c0 3550  {csn 3737  ∪cuni 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892

This theorem is referenced by:  uni0c  3917  uni0  3918