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Theorem abn0r 3220
Description: Nonempty class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
Assertion
Ref Expression
abn0r (xφ → {xφ} ≠ ∅)

Proof of Theorem abn0r
StepHypRef Expression
1 abid 2010 . . 3 (x {xφ} ↔ φ)
21exbii 1478 . 2 (x x {xφ} ↔ xφ)
3 nfab1 2162 . . 3 x{xφ}
43n0rf 3210 . 2 (x x {xφ} → {xφ} ≠ ∅)
52, 4sylbir 125 1 (xφ → {xφ} ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1362   wcel 1374  {cab 2008  wne 2186  c0 3201
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2537  df-dif 2897  df-nul 3202
This theorem is referenced by:  rabn0r  3221
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