Theorem ltpnfd 9977

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 9976	. 2 |- ( A e. RR -> A < +oo )
3	1, 2	syl 14	1 |- ( ph -> A < +oo )

Syntax hints:   -> wi 4   e. wcel 2200   class class class wbr 4083   RRcr 7998   +oocpnf 8178   < clt 8181

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8183  df-xr 8185  df-ltxr 8186

This theorem is referenced by:  xnn0dcle  9998  xqltnle  10487  fprodge1  12150  pcadd  12863