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 2141   ≠ wne 2956  ℝcr 11098  +∞cpnf 11239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-resscn 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-nel 3063  df-rab 3415  df-v 3455  df-in 3911  df-ss 3921  df-pw 4563  df-uni 4872  df-pnf 11244

This theorem is referenced by:  xaddnepnf  13262  dvfsumrlimge0  26168  dvfsumrlim  26169  dvfsumrlim2  26170  logno1  26777  rexmul2  33065  xnn0nn0d  33083  fldextrspundgdvdslem  34036  limsupresico  46384  limsupvaluz2  46422  supcnvlimsup  46424  liminfresico  46455  xlimliminflimsup  46546  smflimsuplem2  47505  smflimsuplem5  47508