Theorem renepnfd 11248

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 11245	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2145   ≠ wne 2960  ℝcr 11087  +∞cpnf 11228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-resscn 11145

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-nel 3065  df-rab 3418  df-v 3459  df-in 3914  df-ss 3924  df-pw 4560  df-uni 4869  df-pnf 11233

This theorem is referenced by:  xaddnepnf  13254  dvfsumrlimge0  26150  dvfsumrlim  26151  dvfsumrlim2  26152  logno1  26759  rexmul2  33011  xnn0nn0d  33029  fldextrspundgdvdslem  33987  limsupresico  46272  limsupvaluz2  46310  supcnvlimsup  46312  liminfresico  46343  xlimliminflimsup  46434  smflimsuplem2  47393  smflimsuplem5  47396