Theorem uni0c 3865

Description: The union of a set is empty iff all of its members are empty. (Contributed by NM, 16-Aug-2006.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem uni0c

Step	Hyp	Ref	Expression
1		uni0b 3864	. 2 ⊢ (∪ 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2		dfss3 3173	. 2 ⊢ (𝐴 ⊆ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ {∅})
3		velsn 3639	. . 3 ⊢ (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
4	3	ralbii 2503	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ {∅} ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)
5	1, 2, 4	3bitri 206	1 ⊢ (∪ 𝐴 = ∅ ↔ ∀𝑥 ∈ 𝐴 𝑥 = ∅)

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  ∅c0 3450  {csn 3622  ∪ cuni 3839

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-uni 3840

This theorem is referenced by: (None)