Theorem ltpnfd 13072

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

Hypothesis

Ref	Expression
ltpnfd.a	⊢ (𝜑 → 𝐴 ∈ ℝ)

Assertion

Ref	Expression
ltpnfd	⊢ (𝜑 → 𝐴 < +∞)

Proof of Theorem ltpnfd

Step	Hyp	Ref	Expression
1		ltpnfd.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		ltpnf 13071	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 < +∞)

Syntax hints:   → wi 4   ∈ wcel 2114   class class class wbr 5085  ℝcr 11037  +∞cpnf 11176   < clt 11179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-pnf 11181  df-xr 11183  df-ltxr 11184

This theorem is referenced by:  qbtwnxr  13152  xltnegi  13168  hashnnn0genn0  14305  limsupgre  15443  fprodge1  15960  xblss2ps  24366  blcvx  24763  reconnlem1  24792  iccpnfhmeo  24912  uniioombllem1  25548  ismbf3d  25621  mbflimsup  25633  itg2seq  25709  lhop2  25982  dvfsumlem2  25994  logccv  26627  xrlimcnp  26932  pntleme  27571  absfico  45647  supxrge  45768  infxr  45796  infleinflem2  45800  xrralrecnnge  45819  iocopn  45950  ge0lere  45962  ressiooinf  45987  uzinico  45989  uzubioo  45995  fsumge0cl  46003  limcicciooub  46065  limcresiooub  46070  limcleqr  46072  limsupresico  46128  limsuppnfdlem  46129  limsupmnflem  46148  liminfresico  46199  limsup10exlem  46200  xlimpnfvlem1  46264  icccncfext  46315  fourierdlem31  46566  fourierdlem33  46568  fourierdlem46  46580  fourierdlem48  46582  fourierdlem49  46583  fourierdlem75  46609  fourierdlem85  46619  fourierdlem88  46622  fourierdlem95  46629  fourierdlem103  46637  fourierdlem104  46638  fourierdlem107  46641  fourierdlem109  46643  fourierdlem112  46646  fouriersw  46659  ioorrnopnxrlem  46734  sge0tsms  46808  sge0isum  46855  sge0ad2en  46859  sge0xaddlem2  46862  voliunsge0lem  46900  meassre  46905  omessre  46938  omeiunltfirp  46947  hoiprodcl  46975  ovnsubaddlem1  46998  hoiprodcl3  47008  hoidmvcl  47010  sge0hsphoire  47017  hoidmv1lelem1  47019  hoidmv1lelem2  47020  hoidmv1lelem3  47021  hoidmv1le  47022  hoidmvlelem1  47023  hoidmvlelem3  47025  hoidmvlelem4  47026  volicorege0  47065  ovolval5lem1  47080