Theorem renepnfd 11223

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 11220	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

Syntax hints:   → wi 4   ∈ wcel 2136   ≠ wne 2951  ℝcr 11062  +∞cpnf 11203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-resscn 11120

This theorem depends on definitions:  df-bi 209  df-an 399  df-3an 1097  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-nel 3056  df-rab 3409  df-v 3450  df-in 3906  df-ss 3916  df-pw 4551  df-uni 4860  df-pnf 11208

This theorem is referenced by:  xaddnepnf  13230  dvfsumrlimge0  26065  dvfsumrlim  26066  dvfsumrlim2  26067  logno1  26671  rexmul2  32899  xnn0nn0d  32917  fldextrspundgdvdslem  33931  limsupresico  46222  limsupvaluz2  46260  supcnvlimsup  46262  liminfresico  46293  xlimliminflimsup  46384  smflimsuplem2  47343  smflimsuplem5  47346