Theorem renepnfd 11196

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 11193	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2114   ≠ wne 2933  ℝcr 11037  +∞cpnf 11176

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 5232  ax-pr 5376  ax-un 7689  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 3391  df-v 3432  df-un 3895  df-in 3897  df-ss 3907  df-pw 4544  df-sn 4569  df-pr 4571  df-uni 4852  df-pnf 11181

This theorem is referenced by:  xaddnepnf  13189  dvfsumrlimge0  25997  dvfsumrlim  25998  dvfsumrlim2  25999  logno1  26600  rexmul2  32827  xnn0nn0d  32845  fldextrspundgdvdslem  33824  limsupresico  46128  limsupvaluz2  46166  supcnvlimsup  46168  liminfresico  46199  xlimliminflimsup  46290  smflimsuplem2  47249  smflimsuplem5  47252