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 in set.mm 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 3579	. 2 |- (/) C_ {(/)}
2		uni0b 3916	. 2 |- (U.(/) = (/) <-> (/) C_ {(/)})
3	1, 2	mpbir 200	1 |- U.(/) = (/)

Syntax hints:   = wceq 1642   C_ wss 3257  (/)c0 3550  {csn 3737  U.cuni 3891

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892

This theorem is referenced by:  uniintsn  3963  iununi  4050  iotanul  4354  dfiota4  4372  funfv  5375