Theorem 9re 9220

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 9199	. 2 ⊢ 9 = (8 + 1)
2		8re 9218	. . 3 ⊢ 8 ∈ ℝ
3		1re 8168	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 8182	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2302	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 2200  (class class class)co 6013  ℝcr 8021  1c1 8023   + caddc 8025  8c8 9190  9c9 9191

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199

This theorem is referenced by:  9cn  9221  7lt9  9332  6lt9  9333  5lt9  9334  4lt9  9335  3lt9  9336  2lt9  9337  1lt9  9338  9lt10  9731  8lt10  9732  0.999...  12072  cos2bnd  12311  sincos2sgn  12317  slotsdifplendx  13283  dsndxntsetndx  13297  unifndxntsetndx  13304  setsmsdsg  15194  2logb9irr  15685  2logb9irrap  15691