Theorem uni0 3688

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	|- U. (/) = (/)

Proof of Theorem uni0

Step	Hyp	Ref	Expression
1		0ss 3327	. 2 |- (/) C_ { (/) }
2		uni0b 3686	. 2 |- ( U. (/) = (/) <-> (/) C_ { (/) } )
3	1, 2	mpbir 145	1 |- U. (/) = (/)

Syntax hints:   = wceq 1290   C_ wss 3002   (/)c0 3289   {csn 3452   U.cuni 3661

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-dif 3004  df-in 3008  df-ss 3015  df-nul 3290  df-sn 3458  df-uni 3662

This theorem is referenced by:  iununir  3820  nnpredcl  4451  unixp0im  4982  iotanul  5010  1st0  5931  2nd0  5932  brtpos0  6033  tpostpos  6045  nnsucuniel  6272  sup00  6754  0opn  11768