Theorem uni0 3771

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 3406	. 2 |- (/) C_ { (/) }
2		uni0b 3769	. 2 |- ( U. (/) = (/) <-> (/) C_ { (/) } )
3	1, 2	mpbir 145	1 |- U. (/) = (/)

Syntax hints:   = wceq 1332   C_ wss 3076   (/)c0 3368   {csn 3532   U.cuni 3744

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369  df-sn 3538  df-uni 3745

This theorem is referenced by:  iununir  3904  nnpredcl  4544  unixp0im  5083  iotanul  5111  1st0  6050  2nd0  6051  brtpos0  6157  tpostpos  6169  nnsucuniel  6399  sup00  6898  0opn  12212