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Theorem 0nn0 7972
Description: 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
Assertion
Ref Expression
0nn0 0 0

Proof of Theorem 0nn0
StepHypRef Expression
1 eqid 2037 . 2 0 = 0
2 elnn0 7959 . . . 4 (0 0 ↔ (0 0 = 0))
32biimpri 124 . . 3 ((0 0 = 0) → 0 0)
43olcs 654 . 2 (0 = 0 → 0 0)
51, 4ax-mp 7 1 0 0
Colors of variables: wff set class
Syntax hints:   wo 628   = wceq 1242   wcel 1390  0cc0 6711  cn 7695  0cn0 7957
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-1cn 6776  ax-icn 6778  ax-addcl 6779  ax-mulcl 6781  ax-i2m1 6788
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-n0 7958
This theorem is referenced by:  elnn0z  8034  nn0ind-raph  8131  numlti  8167  nummul1c  8179  decaddc2  8186  decaddi  8187  decaddci  8188  decaddci2  8189  6p5e11  8193  7p4e11  8195  8p3e11  8199  9p2e11  8205  10p10e20  8213  0elfz  8747  4fvwrd4  8767  exple1  8964
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