Theorem renepnfd 11361

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 11358	. 2 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	1, 2	syl 18	1 ⊢ (𝜑 → 𝐴 ≠ +∞)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144   ≠ wne 2955  ℝcr 11200  +∞cpnf 11341

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-resscn 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-nel 3062  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-pw 4558  df-uni 4867  df-pnf 11346

This theorem is used by:  xaddnepnf  13368  dvfsumrlimge0  26351  dvfsumrlim  26352  dvfsumrlim2  26353  logno1  26965  rexmul2  33389  xnn0nn0d  33407  fldextrspundgdvdslem  34355  limsupresico  46786  limsupvaluz2  46824  supcnvlimsup  46826  liminfresico  46857  xlimliminflimsup  46948  smflimsuplem2  47907  smflimsuplem5  47910