Theorem renepnfd 11195

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 11192	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  ℝcr 11037  +∞cpnf 11175

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 2709  ax-sep 5243  ax-pr 5379  ax-un 7690  ax-resscn 11095

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-nel 3038  df-rab 3402  df-v 3444  df-un 3908  df-in 3910  df-ss 3920  df-pw 4558  df-sn 4583  df-pr 4585  df-uni 4866  df-pnf 11180

This theorem is referenced by:  xaddnepnf  13164  dvfsumrlimge0  26005  dvfsumrlim  26006  dvfsumrlim2  26007  logno1  26613  rexmul2  32844  xnn0nn0d  32862  fldextrspundgdvdslem  33857  limsupresico  46047  limsupvaluz2  46085  supcnvlimsup  46087  liminfresico  46118  xlimliminflimsup  46209  smflimsuplem2  47168  smflimsuplem5  47171