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Theorem nn0nepnf 9195
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 7950 . . . . 5  |- +oo  e/  RR
21neli 2437 . . . 4  |-  -. +oo  e.  RR
3 nn0re 9133 . . . 4  |-  ( +oo  e.  NN0  -> +oo  e.  RR )
42, 3mto 657 . . 3  |-  -. +oo  e.  NN0
5 eleq1 2233 . . 3  |-  ( A  = +oo  ->  ( A  e.  NN0  <-> +oo  e.  NN0 ) )
64, 5mtbiri 670 . 2  |-  ( A  = +oo  ->  -.  A  e.  NN0 )
76necon2ai 2394 1  |-  ( A  e.  NN0  ->  A  =/= +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141    =/= wne 2340   RRcr 7762   +oocpnf 7940   NN0cn0 9124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-un 4416  ax-cnex 7854  ax-resscn 7855  ax-1re 7857  ax-addrcl 7860  ax-rnegex 7872
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-uni 3795  df-int 3830  df-pnf 7945  df-inn 8868  df-n0 9125
This theorem is referenced by:  nn0nepnfd  9197  fxnn0nninf  10383  0tonninf  10384  1tonninf  10385
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