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Theorem uni0c 4430
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 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem uni0c
StepHypRef Expression
1 uni0b 4429 . 2 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2 dfss3 3573 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
3 velsn 4164 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43ralbii 2974 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
51, 2, 43bitri 286 1 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  wral 2907  wss 3555  c0 3891  {csn 4148   cuni 4402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-uni 4403
This theorem is referenced by:  fin1a2lem13  9178  fctop  20718  cctop  20720
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