Theorem dmsnn0 5600

Description: The domain of a singleton is nonzero iff the singleton argument is an ordered pair. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dmsnn0	|- ( A e. ( _V X. _V ) <-> dom { A } =/= (/) )

Proof of Theorem dmsnn0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 |- x e. _V
2	1	eldm 5321	. . . 4 |- ( x e. dom { A } <-> E. y x { A } y )
3		df-br 4654	. . . . . 6 |- ( x { A } y <-> <. x , y >. e. { A } )
4		opex 4932	. . . . . . 7 |- <. x , y >. e. _V
5	4	elsn 4192	. . . . . 6 |- ( <. x , y >. e. { A } <-> <. x , y >. = A )
6		eqcom 2629	. . . . . 6 |- ( <. x , y >. = A <-> A = <. x , y >. )
7	3, 5, 6	3bitri 286	. . . . 5 |- ( x { A } y <-> A = <. x , y >. )
8	7	exbii 1774	. . . 4 |- ( E. y x { A } y <-> E. y A = <. x , y >. )
9	2, 8	bitr2i 265	. . 3 |- ( E. y A = <. x , y >. <-> x e. dom { A } )
10	9	exbii 1774	. 2 |- ( E. x E. y A = <. x , y >. <-> E. x x e. dom { A } )
11		elvv 5177	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
12		n0 3931	. 2 |- ( dom { A } =/= (/) <-> E. x x e. dom { A } )
13	10, 11, 12	3bitr4i 292	1 |- ( A e. ( _V X. _V ) <-> dom { A } =/= (/) )

Syntax hints:   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   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:  rnsnn0  5601  dmsn0  5602  dmsn0el  5604  relsn2  5605  1stnpr  7172  1st2val  7194  mpt2xopxnop0  7341  hashfun  13224