Theorem 9re 9158

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 9137	. 2 ⊢ 9 = (8 + 1)
2		8re 9156	. . 3 ⊢ 8 ∈ ℝ
3		1re 8106	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 8120	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2280	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 2178  (class class class)co 5967  ℝcr 7959  1c1 7961   + caddc 7963  8c8 9128  9c9 9129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-cleq 2200  df-clel 2203  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137

This theorem is referenced by:  9cn  9159  7lt9  9270  6lt9  9271  5lt9  9272  4lt9  9273  3lt9  9274  2lt9  9275  1lt9  9276  9lt10  9669  8lt10  9670  0.999...  11947  cos2bnd  12186  sincos2sgn  12192  slotsdifplendx  13157  dsndxntsetndx  13171  unifndxntsetndx  13178  setsmsdsg  15067  2logb9irr  15558  2logb9irrap  15564