Theorem 9re 9123

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 9102	. 2 ⊢ 9 = (8 + 1)
2		8re 9121	. . 3 ⊢ 8 ∈ ℝ
3		1re 8071	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 8085	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2278	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 2176  (class class class)co 5944  ℝcr 7924  1c1 7926   + caddc 7928  8c8 9093  9c9 9094

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-9 9102

This theorem is referenced by:  9cn  9124  7lt9  9235  6lt9  9236  5lt9  9237  4lt9  9238  3lt9  9239  2lt9  9240  1lt9  9241  9lt10  9634  8lt10  9635  0.999...  11832  cos2bnd  12071  sincos2sgn  12077  slotsdifplendx  13042  dsndxntsetndx  13056  unifndxntsetndx  13063  setsmsdsg  14952  2logb9irr  15443  2logb9irrap  15449