Theorem rexn0 3559

Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3560). (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 3467	. . 3 |- ( x e. A -> A =/= (/) )
2	1	a1d 22	. 2 |- ( x e. A -> ( ph -> A =/= (/) ) )
3	2	rexlimiv 2617	1 |- ( E. x e. A ph -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2176   =/= wne 2376   E.wrex 2485   (/)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-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-nul 3461

This theorem is referenced by: (None)