Theorem n0elqs 34098

Description: Two ways of expressing that the empty set is not an element of a quotient set. (Contributed by Peter Mazsa, 5-Dec-2019.)

Assertion

Ref	Expression
n0elqs	|- ( -. (/) e. ( A /. R ) <-> A C_ dom R )

Proof of Theorem n0elqs

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecdmn0 7789	. . 3 |- ( x e. dom R <-> [ x ] R =/= (/) )
2	1	ralbii 2980	. 2 |- ( A. x e. A x e. dom R <-> A. x e. A [ x ] R =/= (/) )
3		dfss3 3592	. 2 |- ( A C_ dom R <-> A. x e. A x e. dom R )
4		nne 2798	. . . . 5 |- ( -. [ x ] R =/= (/) <-> [ x ] R = (/) )
5	4	rexbii 3041	. . . 4 |- ( E. x e. A -. [ x ] R =/= (/) <-> E. x e. A [ x ] R = (/) )
6	5	notbii 310	. . 3 |- ( -. E. x e. A -. [ x ] R =/= (/) <-> -. E. x e. A [ x ] R = (/) )
7		dfral2 2994	. . 3 |- ( A. x e. A [ x ] R =/= (/) <-> -. E. x e. A -. [ x ] R =/= (/) )
8		0ex 4790	. . . . . 6 |- (/) e. _V
9	8	elqs 7799	. . . . 5 |- ( (/) e. ( A /. R ) <-> E. x e. A (/) = [ x ] R )
10		eqcom 2629	. . . . . 6 |- ( (/) = [ x ] R <-> [ x ] R = (/) )
11	10	rexbii 3041	. . . . 5 |- ( E. x e. A (/) = [ x ] R <-> E. x e. A [ x ] R = (/) )
12	9, 11	bitri 264	. . . 4 |- ( (/) e. ( A /. R ) <-> E. x e. A [ x ] R = (/) )
13	12	notbii 310	. . 3 |- ( -. (/) e. ( A /. R ) <-> -. E. x e. A [ x ] R = (/) )
14	6, 7, 13	3bitr4ri 293	. 2 |- ( -. (/) e. ( A /. R ) <-> A. x e. A [ x ] R =/= (/) )
15	2, 3, 14	3bitr4ri 293	1 |- ( -. (/) e. ( A /. R ) <-> A C_ dom R )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   dom cdm 5114   [cec 7740   /.cqs 7741

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744  df-qs 7748

This theorem is referenced by:  n0elqs2  34099