Theorem 9re 8499

Description: The number 9 is real. (Contributed by NM, 27-May-1999.)

Assertion

Ref	Expression
9re	⊢ 9 ∈ ℝ

Proof of Theorem 9re

Step	Hyp	Ref	Expression
1		df-9 8478	. 2 ⊢ 9 = (8 + 1)
2		8re 8497	. . 3 ⊢ 8 ∈ ℝ
3		1re 7477	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 7491	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2160	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 1438  (class class class)co 5644  ℝcr 7339  1c1 7341   + caddc 7343  8c8 8469  9c9 8470

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-1re 7429  ax-addrcl 7432

This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-clel 2084  df-2 8471  df-3 8472  df-4 8473  df-5 8474  df-6 8475  df-7 8476  df-8 8477  df-9 8478

This theorem is referenced by:  9cn  8500  7lt9  8604  6lt9  8605  5lt9  8606  4lt9  8607  3lt9  8608  2lt9  8609  1lt9  8610  9lt10  8997  8lt10  8998  0.999...  10902  cos2bnd  11038  sincos2sgn  11043