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Theorem nn0nepnf 8680
Description: No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
nn0nepnf  |-  ( A  e.  NN0  ->  A  =/= +oo )

Proof of Theorem nn0nepnf
StepHypRef Expression
1 pnfnre 7476 . . . . 5  |- +oo  e/  RR
21neli 2348 . . . 4  |-  -. +oo  e.  RR
3 nn0re 8618 . . . 4  |-  ( +oo  e.  NN0  -> +oo  e.  RR )
42, 3mto 621 . . 3  |-  -. +oo  e.  NN0
5 eleq1 2147 . . 3  |-  ( A  = +oo  ->  ( A  e.  NN0  <-> +oo  e.  NN0 ) )
64, 5mtbiri 633 . 2  |-  ( A  = +oo  ->  -.  A  e.  NN0 )
76necon2ai 2305 1  |-  ( A  e.  NN0  ->  A  =/= +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436    =/= wne 2251   RRcr 7296   +oocpnf 7466   NN0cn0 8609
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-un 4236  ax-cnex 7383  ax-resscn 7384  ax-1re 7386  ax-addrcl 7389  ax-rnegex 7401
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-uni 3639  df-int 3674  df-pnf 7471  df-inn 8361  df-n0 8610
This theorem is referenced by:  nn0nepnfd  8682  fxnn0nninf  9775  0tonninf  9776  1tonninf  9777
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