Theorem nn0nepnf 9617

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

Step	Hyp	Ref	Expression
1		pnfnre 8357	. . . . 5 |- +oo e/ RR
2	1	neli 2517	. . . 4 |- -. +oo e. RR
3		nn0re 9551	. . . 4 |- ( +oo e. NN0 -> +oo e. RR )
4	2, 3	mto 672	. . 3 |- -. +oo e. NN0
5		eleq1 2301	. . 3 |- ( A = +oo -> ( A e. NN0 <-> +oo e. NN0 ) )
6	4, 5	mtbiri 686	. 2 |- ( A = +oo -> -. A e. NN0 )
7	6	necon2ai 2474	1 |- ( A e. NN0 -> A =/= +oo )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   =/= wne 2420   RRcr 8168   +oocpnf 8347   NN0cn0 9542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-rnegex 8278

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-uni 3931  df-int 3966  df-pnf 8352  df-inn 9284  df-n0 9543

This theorem is referenced by:  nn0nepnfd  9619  xnn0nnen  10852  fxnn0nninf  10854  0tonninf  10855  1tonninf  10856