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Theorem uni0c 3847
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 3846 . 2 ( 𝐴 = ∅ ↔ 𝐴 ⊆ {∅})
2 dfss3 3157 . 2 (𝐴 ⊆ {∅} ↔ ∀𝑥𝐴 𝑥 ∈ {∅})
3 velsn 3621 . . 3 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
43ralbii 2493 . 2 (∀𝑥𝐴 𝑥 ∈ {∅} ↔ ∀𝑥𝐴 𝑥 = ∅)
51, 2, 43bitri 206 1 ( 𝐴 = ∅ ↔ ∀𝑥𝐴 𝑥 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1363  wcel 2158  wral 2465  wss 3141  c0 3434  {csn 3604   cuni 3821
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-uni 3822
This theorem is referenced by: (None)
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