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Theorem rexn0 3319
 Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3320). (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 3230 . . 3 (𝑥𝐴𝐴 ≠ ∅)
21a1d 22 . 2 (𝑥𝐴 → (𝜑𝐴 ≠ ∅))
32rexlimiv 2427 1 (∃𝑥𝐴 𝜑𝐴 ≠ ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393   ≠ wne 2204  ∃wrex 2307  ∅c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-nul 3225 This theorem is referenced by: (None)
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