Theorem dom0 7023

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 6919	. . 3 |- ( A ~<_ (/) -> E. x x : A -1-1-> (/) )
2		f1f 5542	. . . . . 6 |- ( x : A -1-1-> (/) -> x : A --> (/) )
3		f00 5528	. . . . . 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 1947	. 2 |- ( A ~<_ (/) -> A = (/) )
8		0ex 4216	. . . 4 |- (/) e. _V
9		domrefg 6939	. . . 4 |- ( (/) e. _V -> (/) ~<_ (/) )
10	8, 9	ax-mp 5	. . 3 |- (/) ~<_ (/)
11		breq1 4091	. . 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 1397   e. wcel 2202   _Vcvv 2802   (/)c0 3494   class class class wbr 4088   -->wf 5322   -1-1->wf1 5323   ~<_ cdom 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-en 6909  df-dom 6910

This theorem is referenced by: (None)