Theorem elqsn0 6772

Description: A quotient set doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

Assertion

Ref	Expression
elqsn0	|- ( ( dom R = A /\ B e. ( A /. R ) ) -> B =/= (/) )

Proof of Theorem elqsn0

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsn0m 6771	. 2 |- ( ( dom R = A /\ B e. ( A /. R ) ) -> E. x x e. B )
2		n0r 3508	. 2 |- ( E. x x e. B -> B =/= (/) )
3	1, 2	syl 14	1 |- ( ( dom R = A /\ B e. ( A /. R ) ) -> B =/= (/) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   =/= wne 2402   (/)c0 3494   dom cdm 4725   /.cqs 6700

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-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

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-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  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-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-ec 6703  df-qs 6707

This theorem is referenced by:  0nnq  7583  0nsr  7968