Theorem dmsn0 5602

Description: The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.)

Assertion

Ref	Expression
dmsn0	|- dom { (/) } = (/)

Proof of Theorem dmsn0

Step	Hyp	Ref	Expression
1		0nelxp 5143	. 2 |- -. (/) e. ( _V X. _V )
2		dmsnn0 5600	. . 3 |- ( (/) e. ( _V X. _V ) <-> dom { (/) } =/= (/) )
3	2	necon2bbii 2845	. 2 |- ( dom { (/) } = (/) <-> -. (/) e. ( _V X. _V ) )
4	1, 3	mpbir 221	1 |- dom { (/) } = (/)

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   X. cxp 5112   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-dm 5124

This theorem is referenced by:  cnvsn0  5603  dmsnopss  5607  1st0  7174  2nd0  7175