Theorem uni0 3863

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 3486	. 2 |- (/) C_ { (/) }
2		uni0b 3861	. 2 |- ( U. (/) = (/) <-> (/) C_ { (/) } )
3	1, 2	mpbir 146	1 |- U. (/) = (/)

Syntax hints:   = wceq 1364   C_ wss 3154   (/)c0 3447   {csn 3619   U.cuni 3836

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-in 3160  df-ss 3167  df-nul 3448  df-sn 3625  df-uni 3837

This theorem is referenced by:  iununir  3997  nnpredcl  4656  unixp0im  5203  iotanul  5231  1st0  6199  2nd0  6200  brtpos0  6307  tpostpos  6319  nnsucuniel  6550  sup00  7064  nnnninfeq2  7190  0opn  14185