Theorem 9re 9370

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 9349	. 2 ⊢ 9 = (8 + 1)
2		8re 9368	. . 3 ⊢ 8 ∈ ℝ
3		1re 8315	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 8329	. 2 ⊢ (8 + 1) ∈ ℝ
5	1, 4	eqeltri 2311	1 ⊢ 9 ∈ ℝ

Syntax hints:   ∈ wcel 2209  (class class class)co 6075  ℝcr 8168  1c1 8170   + caddc 8172  8c8 9340  9c9 9341

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349

This theorem is referenced by:  9cn  9371  7lt9  9482  6lt9  9483  5lt9  9484  4lt9  9485  3lt9  9486  2lt9  9487  1lt9  9488  9lt10  9886  8lt10  9887  0.999...  12266  cos2bnd  12505  sincos2sgn  12511  slotsdifplendx  13541  dsndxntsetndx  13555  unifndxntsetndx  13562  setsmsdsg  15504  2logb9irr  15996  2logb9irrap  16002