Theorem ltpnfd 10165

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 10164	. 2 |- ( A e. RR -> A < +oo )
3	1, 2	syl 14	1 |- ( ph -> A < +oo )

Syntax hints:   -> wi 4   e. wcel 2209   class class class wbr 4128   RRcr 8171   +oocpnf 8350   < clt 8353

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 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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8263

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-pnf 8355  df-xr 8357  df-ltxr 8358

This theorem is referenced by:  xnn0dcle  10186  xqltnle  10683  fprodge1  12387  pcadd  13100  repiecelem  16982  repiecele0  16983  repiecege0  16984