Theorem renepnfd 11166

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 11163	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2109   ≠ wne 2925  ℝcr 11008  +∞cpnf 11146

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-pr 5371  ax-un 7671  ax-resscn 11066

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-nel 3030  df-rab 3395  df-v 3438  df-un 3908  df-in 3910  df-ss 3920  df-pw 4553  df-sn 4578  df-pr 4580  df-uni 4859  df-pnf 11151

This theorem is referenced by:  xaddnepnf  13139  dvfsumrlimge0  25935  dvfsumrlim  25936  dvfsumrlim2  25937  logno1  26543  rexmul2  32706  xnn0nn0d  32724  fldextrspundgdvdslem  33663  limsupresico  45701  limsupvaluz2  45739  supcnvlimsup  45741  liminfresico  45772  xlimliminflimsup  45863  smflimsuplem2  46822  smflimsuplem5  46825