Theorem uni0b 1939

Description: The union of a set is empty iff the set is included in the singleton of the empty set.

Assertion

Ref	Expression
uni0b	⊢ (∪A = ∅ ↔ A ⊆ {∅})

Proof of Theorem uni0b

Step	Hyp	Ref	Expression
1		elsn 1820	. . 3 ⊢ (x ∈ {∅} ↔ x = ∅)
2	1	biral 1223	. 2 ⊢ (∀x ∈ A x ∈ {∅} ↔ ∀x ∈ A x = ∅)
3		dfss3 1498	. 2 ⊢ (A ⊆ {∅} ↔ ∀x ∈ A x ∈ {∅})
4		n0 1714	. . . 4 ⊢ (¬ ∪A = ∅ ↔ ∃y y ∈ ∪A)
5		rexcom4 1361	. . . . 5 ⊢ (∃x ∈ A ∃y y ∈ x ↔ ∃y∃x ∈ A y ∈ x)
6		n0 1714	. . . . . 6 ⊢ (¬ x = ∅ ↔ ∃y y ∈ x)
7	6	birex 1224	. . . . 5 ⊢ (∃x ∈ A ¬ x = ∅ ↔ ∃x ∈ A ∃y y ∈ x)
8		eluni2 1923	. . . . . 6 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
9	8	biex 733	. . . . 5 ⊢ (∃y y ∈ ∪A ↔ ∃y∃x ∈ A y ∈ x)
10	5, 7, 9	3bitr4r 159	. . . 4 ⊢ (∃y y ∈ ∪A ↔ ∃x ∈ A ¬ x = ∅)
11		rexnal 1210	. . . 4 ⊢ (∃x ∈ A ¬ x = ∅ ↔ ¬ ∀x ∈ A x = ∅)
12	4, 10, 11	3bitr 155	. . 3 ⊢ (¬ ∪A = ∅ ↔ ¬ ∀x ∈ A x = ∅)
13	12	bicon4i 401	. 2 ⊢ (∪A = ∅ ↔ ∀x ∈ A x = ∅)
14	2, 3, 13	3bitr4r 159	1 ⊢ (∪A = ∅ ↔ A ⊆ {∅})

Syntax hints:  ¬ wn 1   ↔ wb 127  ∃wex 678   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  ∅c0 1707  {csn 1808  ∪cuni 1919

This theorem is referenced by:  infxpidmlem8 4940

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-uni 1920

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