Theorem nn0nepnf 9517

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 8263	. . . . 5 |- +oo e/ RR
2	1	neli 2500	. . . 4 |- -. +oo e. RR
3		nn0re 9453	. . . 4 |- ( +oo e. NN0 -> +oo e. RR )
4	2, 3	mto 668	. . 3 |- -. +oo e. NN0
5		eleq1 2294	. . 3 |- ( A = +oo -> ( A e. NN0 <-> +oo e. NN0 ) )
6	4, 5	mtbiri 682	. 2 |- ( A = +oo -> -. A e. NN0 )
7	6	necon2ai 2457	1 |- ( A e. NN0 -> A =/= +oo )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   RRcr 8074   +oocpnf 8253   NN0cn0 9444

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172  ax-rnegex 8184

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-uni 3899  df-int 3934  df-pnf 8258  df-inn 9186  df-n0 9445

This theorem is referenced by:  nn0nepnfd  9519  xnn0nnen  10745  fxnn0nninf  10747  0tonninf  10748  1tonninf  10749