Theorem uni0 3918

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 3531	. 2 |- (/) C_ { (/) }
2		uni0b 3916	. 2 |- ( U. (/) = (/) <-> (/) C_ { (/) } )
3	1, 2	mpbir 146	1 |- U. (/) = (/)

Syntax hints:   = wceq 1395   C_ wss 3198   (/)c0 3492   {csn 3667   U.cuni 3891

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-in 3204  df-ss 3211  df-nul 3493  df-sn 3673  df-uni 3892

This theorem is referenced by:  iununir  4052  nnpredcl  4719  unixp0im  5271  iotanul  5300  1st0  6302  2nd0  6303  brtpos0  6413  tpostpos  6425  nnsucuniel  6658  sup00  7193  nnnninfeq2  7319  0opn  14720