Theorem renepnfd 11185

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 11182	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2931  ℝcr 11027  +∞cpnf 11165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-pr 5376  ax-un 7680  ax-resscn 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-nel 3036  df-rab 3399  df-v 3441  df-un 3905  df-in 3907  df-ss 3917  df-pw 4555  df-sn 4580  df-pr 4582  df-uni 4863  df-pnf 11170

This theorem is referenced by:  xaddnepnf  13154  dvfsumrlimge0  25995  dvfsumrlim  25996  dvfsumrlim2  25997  logno1  26603  rexmul2  32813  xnn0nn0d  32831  fldextrspundgdvdslem  33816  limsupresico  45981  limsupvaluz2  46019  supcnvlimsup  46021  liminfresico  46052  xlimliminflimsup  46143  smflimsuplem2  47102  smflimsuplem5  47105