Theorem uni0c 2528

Description: The union of a set is empty iff all of its members are empty.

Assertion

Ref	Expression
uni0c	⊢ (∪A = ∅ ↔ ∀x ∈ A x = ∅)

Distinct variable group:   x,A

Proof of Theorem uni0c

Step	Hyp	Ref	Expression
1		uni0b 2527	. 2 ⊢ (∪A = ∅ ↔ A ⊆ {∅})
2		dfss3 2062	. 2 ⊢ (A ⊆ {∅} ↔ ∀x ∈ A x ∈ {∅})
3		elsn 2425	. . 3 ⊢ (x ∈ {∅} ↔ x = ∅)
4	3	ralbii 1670	. 2 ⊢ (∀x ∈ A x ∈ {∅} ↔ ∀x ∈ A x = ∅)
5	1, 2, 4	3bitr 177	1 ⊢ (∪A = ∅ ↔ ∀x ∈ A x = ∅)

Syntax hints:   ↔ wb 146   = wceq 958   ∈ wcel 960  ∀wral 1648   ⊆ wss 2050  ∅c0 2283  {csn 2413  ∪cuni 2507

This theorem is referenced by:  fctopOLD 7647

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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