Theorem 9re 9197

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 9176	. 2 ⊢ 9 = (8 + 1)
2		8re 9195	. . 3 ⊢ 8 ∈ ℝ
3		1re 8145	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 8159	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2302	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 2200  (class class class)co 6001  ℝcr 7998  1c1 8000   + caddc 8002  8c8 9167  9c9 9168

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 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176

This theorem is referenced by:  9cn  9198  7lt9  9309  6lt9  9310  5lt9  9311  4lt9  9312  3lt9  9313  2lt9  9314  1lt9  9315  9lt10  9708  8lt10  9709  0.999...  12032  cos2bnd  12271  sincos2sgn  12277  slotsdifplendx  13243  dsndxntsetndx  13257  unifndxntsetndx  13264  setsmsdsg  15154  2logb9irr  15645  2logb9irrap  15651