Theorem 9re 9122

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 9101	. 2 ⊢ 9 = (8 + 1)
2		8re 9120	. . 3 ⊢ 8 ∈ ℝ
3		1re 8070	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 8084	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2277	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 2175  (class class class)co 5943  ℝcr 7923  1c1 7925   + caddc 7927  8c8 9092  9c9 9093

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-clel 2200  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-6 9098  df-7 9099  df-8 9100  df-9 9101

This theorem is referenced by:  9cn  9123  7lt9  9234  6lt9  9235  5lt9  9236  4lt9  9237  3lt9  9238  2lt9  9239  1lt9  9240  9lt10  9633  8lt10  9634  0.999...  11803  cos2bnd  12042  sincos2sgn  12048  slotsdifplendx  13013  dsndxntsetndx  13027  unifndxntsetndx  13034  setsmsdsg  14923  2logb9irr  15414  2logb9irrap  15420