Theorem renepnfd 11259

Description: No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Hypothesis

Ref	Expression
rexrd.1	⊢ (𝜑 → 𝐴 ∈ ℝ)

Assertion

Ref	Expression
renepnfd	⊢ (𝜑 → 𝐴 ≠ +∞)

Proof of Theorem renepnfd

Step	Hyp	Ref	Expression
1		rexrd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		renepnf 11256	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2149   ≠ wne 2964  ℝcr 11098  +∞cpnf 11239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-resscn 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-nel 3071  df-rab 3424  df-v 3465  df-in 3920  df-ss 3930  df-pw 4569  df-uni 4877  df-pnf 11244

This theorem is referenced by:  xaddnepnf  13262  dvfsumrlimge0  26157  dvfsumrlim  26158  dvfsumrlim2  26159  logno1  26766  rexmul2  33039  xnn0nn0d  33057  fldextrspundgdvdslem  34014  limsupresico  46305  limsupvaluz2  46343  supcnvlimsup  46345  liminfresico  46376  xlimliminflimsup  46467  smflimsuplem2  47426  smflimsuplem5  47429