Theorem dom0 7093

Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)

Assertion

Ref	Expression
dom0	|- ( A ~<_ (/) <-> A = (/) )

Proof of Theorem dom0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 6988	. . 3 |- ( A ~<_ (/) -> E. x x : A -1-1-> (/) )
2		f1f 5575	. . . . . 6 |- ( x : A -1-1-> (/) -> x : A --> (/) )
3		f00 5561	. . . . . 6 |- ( x : A --> (/) <-> ( x = (/) /\ A = (/) ) )
4	2, 3	sylib 122	. . . . 5 |- ( x : A -1-1-> (/) -> ( x = (/) /\ A = (/) ) )
5	4	simprd 114	. . . 4 |- ( x : A -1-1-> (/) -> A = (/) )
6	5	adantl 277	. . 3 |- ( ( A ~<_ (/) /\ x : A -1-1-> (/) ) -> A = (/) )
7	1, 6	exlimddv 1950	. 2 |- ( A ~<_ (/) -> A = (/) )
8		0ex 4239	. . . 4 |- (/) e. _V
9		domrefg 7008	. . . 4 |- ( (/) e. _V -> (/) ~<_ (/) )
10	8, 9	ax-mp 5	. . 3 |- (/) ~<_ (/)
11		breq1 4114	. . 3 |- ( A = (/) -> ( A ~<_ (/) <-> (/) ~<_ (/) ) )
12	10, 11	mpbiri 168	. 2 |- ( A = (/) -> A ~<_ (/) )
13	7, 12	impbii 126	1 |- ( A ~<_ (/) <-> A = (/) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   _Vcvv 2815   (/)c0 3510   class class class wbr 4111   -->wf 5350   -1-1->wf1 5351   ~<_ cdom 6976

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-en 6978  df-dom 6979

This theorem is referenced by: (None)