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Theorem rexn0 3427
Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3428). (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
StepHypRef Expression
1 ne0i 3335 . . 3  |-  ( x  e.  A  ->  A  =/=  (/) )
21a1d 22 . 2  |-  ( x  e.  A  ->  ( ph  ->  A  =/=  (/) ) )
32rexlimiv 2517 1  |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1463    =/= wne 2282   E.wrex 2391   (/)c0 3329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-v 2659  df-dif 3039  df-nul 3330
This theorem is referenced by: (None)
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