Theorem renepnfd 11287

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 11284	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ≠ wne 2957  ℝcr 11126  +∞cpnf 11267

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 2734  ax-sep 5255  ax-resscn 11184

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 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-nel 3064  df-rab 3415  df-v 3455  df-in 3909  df-ss 3919  df-pw 4562  df-uni 4871  df-pnf 11272

This theorem is used by:  xaddnepnf  13291  dvfsumrlimge0  26262  dvfsumrlim  26263  dvfsumrlim2  26264  logno1  26874  rexmul2  33227  xnn0nn0d  33245  fldextrspundgdvdslem  34192  limsupresico  46530  limsupvaluz2  46568  supcnvlimsup  46570  liminfresico  46601  xlimliminflimsup  46692  smflimsuplem2  47651  smflimsuplem5  47654