Theorem renepnfd 11194

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 11191	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2119   ≠ wne 2935  ℝcr 11035  +∞cpnf 11174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369  ax-un 7685  ax-resscn 11093

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-nel 3040  df-rab 3393  df-v 3434  df-un 3895  df-in 3897  df-ss 3907  df-pw 4538  df-sn 4563  df-pr 4565  df-uni 4846  df-pnf 11179

This theorem is referenced by:  xaddnepnf  13187  dvfsumrlimge0  26022  dvfsumrlim  26023  dvfsumrlim2  26024  logno1  26625  rexmul2  32853  xnn0nn0d  32871  fldextrspundgdvdslem  33871  limsupresico  46144  limsupvaluz2  46182  supcnvlimsup  46184  liminfresico  46215  xlimliminflimsup  46306  smflimsuplem2  47265  smflimsuplem5  47268