Theorem eq0rdv 3505

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 620	. . 3 |- ( ph -> ( x e. A -> x e. (/) ) )
3	2	ssrdv 3199	. 2 |- ( ph -> A C_ (/) )
4		ss0 3501	. 2 |- ( A C_ (/) -> A = (/) )
5	3, 4	syl 14	1 |- ( ph -> A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1373   e. wcel 2176   C_ wss 3166   (/)c0 3460

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by:  exmid01  4242  dcextest  4629  nfvres  5610  map0b  6774  snon0  7037  snexxph  7052  fodju0  7249  fzdisj  10174  bldisj  14873