Theorem eq0rdv 3539

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 624	. . 3 |- ( ph -> ( x e. A -> x e. (/) ) )
3	2	ssrdv 3233	. 2 |- ( ph -> A C_ (/) )
4		ss0 3535	. 2 |- ( A C_ (/) -> A = (/) )
5	3, 4	syl 14	1 |- ( ph -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   e. wcel 2202   C_ wss 3200   (/)c0 3494

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-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495

This theorem is referenced by:  exmid01  4288  dcextest  4679  nfvres  5675  map0b  6855  snon0  7133  snexxph  7148  fodju0  7345  fzdisj  10286  bldisj  15124  usgr1vr  16098