Theorem eq0rdv 3570

Description: Deduction 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 628	. . 3 |- ( ph -> ( x e. A -> x e. (/) ) )
3	2	ssrdv 3254	. 2 |- ( ph -> A C_ (/) )
4		ss0 3563	. 2 |- ( A C_ (/) -> A = (/) )
5	3, 4	syl 14	1 |- ( ph -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1402   e. wcel 2209   C_ wss 3220   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by:  exmid01  4330  dcextest  4723  nfvres  5726  map0b  6958  snon0  7239  snexxph  7257  fodju0  7477  fzdisj  10435  bldisj  15425  usgr1vr  16403