Theorem uni0b 2527

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 2425	. . 3 ⊢ (x ∈ {∅} ↔ x = ∅)
2	1	ralbii 1670	. 2 ⊢ (∀x ∈ A x ∈ {∅} ↔ ∀x ∈ A x = ∅)
3		dfss3 2062	. 2 ⊢ (A ⊆ {∅} ↔ ∀x ∈ A x ∈ {∅})
4		n0 2293	. . . 4 ⊢ (¬ ∪A = ∅ ↔ ∃y y ∈ ∪A)
5		rexcom4 1827	. . . . 5 ⊢ (∃x ∈ A ∃y y ∈ x ↔ ∃y∃x ∈ A y ∈ x)
6		n0 2293	. . . . . 6 ⊢ (¬ x = ∅ ↔ ∃y y ∈ x)
7	6	rexbii 1671	. . . . 5 ⊢ (∃x ∈ A ¬ x = ∅ ↔ ∃x ∈ A ∃y y ∈ x)
8		eluni2 2511	. . . . . 6 ⊢ (y ∈ ∪A ↔ ∃x ∈ A y ∈ x)
9	8	exbii 1053	. . . . 5 ⊢ (∃y y ∈ ∪A ↔ ∃y∃x ∈ A y ∈ x)
10	5, 7, 9	3bitr4r 184	. . . 4 ⊢ (∃y y ∈ ∪A ↔ ∃x ∈ A ¬ x = ∅)
11		rexnal 1657	. . . 4 ⊢ (∃x ∈ A ¬ x = ∅ ↔ ¬ ∀x ∈ A x = ∅)
12	4, 10, 11	3bitr 177	. . 3 ⊢ (¬ ∪A = ∅ ↔ ¬ ∀x ∈ A x = ∅)
13	12	con4bii 525	. 2 ⊢ (∪A = ∅ ↔ ∀x ∈ A x = ∅)
14	2, 3, 13	3bitr4r 184	1 ⊢ (∪A = ∅ ↔ A ⊆ {∅})

Syntax hints:  ¬ wn 2   ↔ wb 146   = wceq 958   ∈ wcel 960  ∃wex 982  ∀wral 1648  ∃wrex 1649   ⊆ wss 2050  ∅c0 2283  {csn 2413  ∪cuni 2507

This theorem is referenced by:  uni0c 2528  uni0 2529  infxpidmlem8 7560  0top 7634  cctop 7649  top2usne 10535

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-uni 2508

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