Theorem ltpnfd 10077

Description: Any (finite) real is less than plus infinity. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
ltpnfd.a	|- ( ph -> A e. RR )

Assertion

Ref	Expression
ltpnfd	|- ( ph -> A < +oo )

Proof of Theorem ltpnfd

Step	Hyp	Ref	Expression
1		ltpnfd.a	. 2 |- ( ph -> A e. RR )
2		ltpnf 10076	. 2 |- ( A e. RR -> A < +oo )
3	1, 2	syl 14	1 |- ( ph -> A < +oo )

Syntax hints:   -> wi 4   e. wcel 2202   class class class wbr 4093   RRcr 8091   +oocpnf 8270   < clt 8273

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-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-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8183

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-pnf 8275  df-xr 8277  df-ltxr 8278

This theorem is referenced by:  xnn0dcle  10098  xqltnle  10590  fprodge1  12280  pcadd  12993  repiecelem  16757  repiecele0  16758  repiecege0  16759