Theorem renepnfd 11266

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 11263	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   ≠ wne 2957  ℝcr 11105  +∞cpnf 11246

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-resscn 11163

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064  df-rab 3416  df-v 3456  df-in 3911  df-ss 3921  df-pw 4563  df-uni 4872  df-pnf 11251

This theorem is used by:  xaddnepnf  13269  dvfsumrlimge0  26200  dvfsumrlim  26201  dvfsumrlim2  26202  logno1  26812  rexmul2  33110  xnn0nn0d  33128  fldextrspundgdvdslem  34079  limsupresico  46442  limsupvaluz2  46480  supcnvlimsup  46482  liminfresico  46513  xlimliminflimsup  46604  smflimsuplem2  47563  smflimsuplem5  47566