Theorem 8re 8829

Description: The number 8 is real. (Contributed by NM, 27-May-1999.)

Assertion

Ref	Expression
8re	⊢ 8 ∈ ℝ

Proof of Theorem 8re

Step	Hyp	Ref	Expression
1		df-8 8809	. 2 ⊢ 8 = (7 + 1)
2		7re 8827	. . 3 ⊢ 7 ∈ ℝ
3		1re 7789	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 7803	. 2 ⊢ (7 + 1) ∈ ℝ
5	1, 4	eqeltri 2213	1 ⊢ 8 ∈ ℝ

Syntax hints:   ∈ wcel 1481  (class class class)co 5782  ℝcr 7643  1c1 7645   + caddc 7647  7c7 8800  8c8 8801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122  ax-1re 7738  ax-addrcl 7741

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136  df-2 8803  df-3 8804  df-4 8805  df-5 8806  df-6 8807  df-7 8808  df-8 8809

This theorem is referenced by:  8cn  8830  9re  8831  9pos  8848  6lt8  8935  5lt8  8936  4lt8  8937  3lt8  8938  2lt8  8939  1lt8  8940  8lt9  8941  7lt9  8942  8th4div3  8963  8lt10  9337  7lt10  9338  ef01bndlem  11499  cos2bnd  11503