Theorem rexn0 4074

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Assertion

Ref	Expression
rexn0	|- ( E. x e. A ph -> A =/= (/) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rexn0

Step	Hyp	Ref	Expression
1		ne0i 3921	. . 3 |- ( x e. A -> A =/= (/) )
2	1	a1d 25	. 2 |- ( x e. A -> ( ph -> A =/= (/) ) )
3	2	rexlimiv 3027	1 |- ( E. x e. A ph -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1990   =/= wne 2794   E.wrex 2913   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  reusv2lem3  4871  eusvobj2  6643  isdrs2  16939  ismnd  17297  slwn0  18030  lbsexg  19164  iunconn  21231  grpon0  27356  filbcmb  33535  isbnd2  33582  rencldnfi  37385  iunconnlem2  39171  stoweidlem14  40231  hoidmvval0  40801  2reu4  41190