Theorem renepnfd 11332

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 11329	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2955  ℝcr 11171  +∞cpnf 11312

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-resscn 11229

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-nel 3062  df-rab 3413  df-v 3452  df-in 3905  df-ss 3915  df-pw 4558  df-uni 4867  df-pnf 11317

This theorem is used by:  xaddnepnf  13337  dvfsumrlimge0  26312  dvfsumrlim  26313  dvfsumrlim2  26314  logno1  26928  rexmul2  33280  xnn0nn0d  33298  fldextrspundgdvdslem  34246  limsupresico  46632  limsupvaluz2  46670  supcnvlimsup  46672  liminfresico  46703  xlimliminflimsup  46794  smflimsuplem2  47753  smflimsuplem5  47756