Theorem uni0 3891

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 3507	. 2 |- (/) C_ { (/) }
2		uni0b 3889	. 2 |- ( U. (/) = (/) <-> (/) C_ { (/) } )
3	1, 2	mpbir 146	1 |- U. (/) = (/)

Syntax hints:   = wceq 1373   C_ wss 3174   (/)c0 3468   {csn 3643   U.cuni 3864

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649  df-uni 3865

This theorem is referenced by:  iununir  4025  nnpredcl  4689  unixp0im  5238  iotanul  5266  1st0  6253  2nd0  6254  brtpos0  6361  tpostpos  6373  nnsucuniel  6604  sup00  7131  nnnninfeq2  7257  0opn  14593