Theorem renepnfd 11264

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 11261	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2106   ≠ wne 2940  ℝcr 11108  +∞cpnf 11244

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-pr 5427  ax-un 7724  ax-resscn 11166

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-nel 3047  df-rab 3433  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-pw 4604  df-sn 4629  df-pr 4631  df-uni 4909  df-pnf 11249

This theorem is referenced by:  xaddnepnf  13215  dvfsumrlimge0  25546  dvfsumrlim  25547  dvfsumrlim2  25548  logno1  26143  limsupresico  44406  limsupvaluz2  44444  supcnvlimsup  44446  liminfresico  44477  xlimliminflimsup  44568  smflimsuplem2  45527  smflimsuplem5  45530