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Theorem uni0 3821
Description: The union of the empty set is the empty set. Theorem 8.7 of [Quine] p. 54. (Reproved without relying on ax-nul by Eric Schmidt.) (Contributed by NM, 16-Sep-1993.) (Revised by Eric Schmidt, 4-Apr-2007.)
Assertion
Ref Expression
uni0 ∅ = ∅

Proof of Theorem uni0
StepHypRef Expression
1 0ss 3452 . 2 ∅ ⊆ {∅}
2 uni0b 3819 . 2 ( ∅ = ∅ ↔ ∅ ⊆ {∅})
31, 2mpbir 145 1 ∅ = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1348  wss 3121  c0 3414  {csn 3581   cuni 3794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3587  df-uni 3795
This theorem is referenced by:  iununir  3954  nnpredcl  4605  unixp0im  5145  iotanul  5173  1st0  6120  2nd0  6121  brtpos0  6228  tpostpos  6240  nnsucuniel  6471  sup00  6976  nnnninfeq2  7101  0opn  12757
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