Theorem uni0 3837

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 3462	. 2 |- (/) C_ { (/) }
2		uni0b 3835	. 2 |- ( U. (/) = (/) <-> (/) C_ { (/) } )
3	1, 2	mpbir 146	1 |- U. (/) = (/)

Syntax hints:   = wceq 1353   C_ wss 3130   (/)c0 3423   {csn 3593   U.cuni 3810

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-in 3136  df-ss 3143  df-nul 3424  df-sn 3599  df-uni 3811

This theorem is referenced by:  iununir  3971  nnpredcl  4623  unixp0im  5166  iotanul  5194  1st0  6145  2nd0  6146  brtpos0  6253  tpostpos  6265  nnsucuniel  6496  sup00  7002  nnnninfeq2  7127  0opn  13509