Theorem eq0rdv 3261

Description: Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)

Hypothesis

Ref	Expression
eq0rdv.1	|- ( ph -> -. x e. A )

Assertion

Ref	Expression
eq0rdv	|- ( ph -> A = (/) )

Distinct variable groups:   x, A   ph, x

Proof of Theorem eq0rdv

Step	Hyp	Ref	Expression
1		eq0rdv.1	. . . 4 |- ( ph -> -. x e. A )
2	1	pm2.21d 549	. . 3 |- ( ph -> ( x e. A -> x e. (/) ) )
3	2	ssrdv 2951	. 2 |- ( ph -> A C_ (/) )
4		ss0 3257	. 2 |- ( A C_ (/) -> A = (/) )
5	3, 4	syl 14	1 |- ( ph -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1243   e. wcel 1393   C_ wss 2917   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225

This theorem is referenced by:  nfvres  5206  snon0  6356  fzdisj  8916