Theorem uni0 3657

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 3306	. 2 |- (/) C_ { (/) }
2		uni0b 3655	. 2 |- ( U. (/) = (/) <-> (/) C_ { (/) } )
3	1, 2	mpbir 144	1 |- U. (/) = (/)

Syntax hints:   = wceq 1287   C_ wss 2986   (/)c0 3272   {csn 3425   U.cuni 3630

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-dif 2988  df-in 2992  df-ss 2999  df-nul 3273  df-sn 3431  df-uni 3631

This theorem is referenced by:  iununir  3788  unixp0im  4924  iotanul  4952  1st0  5853  2nd0  5854  brtpos0  5952  tpostpos  5964  nnsucuniel  6191  sup00  6619  nnpredcl  11246