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Theorem rexn0 3347
Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3348). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
Assertion
Ref Expression
rexn0 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexn0
StepHypRef Expression
1 ne0i 3264 . . 3 (𝑥𝐴𝐴 ≠ ∅)
21a1d 22 . 2 (𝑥𝐴 → (𝜑𝐴 ≠ ∅))
32rexlimiv 2472 1 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wne 2246  wrex 2350  c0 3258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-nul 3259
This theorem is referenced by: (None)
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