Theorem uni0 3962

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 3561	. 2 |- (/) C_ { (/) }
2		uni0b 3960	. 2 |- ( U. (/) = (/) <-> (/) C_ { (/) } )
3	1, 2	mpbir 146	1 |- U. (/) = (/)

This proof depends on syntax axioms:   = wceq 1402   C_ wss 3220   (/)c0 3520   {csn 3709   U.cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3715  df-uni 3936

This theorem is used by:  iununir  4096  nnpredcl  4770  unixp0im  5324  iotanul  5353  1st0  6378  2nd0  6379  brtpos0  6523  tpostpos  6535  nnsucuniel  6768  sup00  7343  nnnninfeq2  7469  0opn  15107