Theorem 8re 8963

Description: The number 8 is real. (Contributed by NM, 27-May-1999.)

Assertion

Ref	Expression
8re	⊢ 8 ∈ ℝ

Proof of Theorem 8re

Step	Hyp	Ref	Expression
1		df-8 8943	. 2 ⊢ 8 = (7 + 1)
2		7re 8961	. . 3 ⊢ 7 ∈ ℝ
3		1re 7919	. . 3 ⊢ 1 ∈ ℝ
4	2, 3	readdcli 7933	. 2 ⊢ (7 + 1) ∈ ℝ
5	1, 4	eqeltri 2243	1 ⊢ 8 ∈ ℝ

Syntax hints:   ∈ wcel 2141  (class class class)co 5853  ℝcr 7773  1c1 7775   + caddc 7777  7c7 8934  8c8 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152  ax-1re 7868  ax-addrcl 7871

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943

This theorem is referenced by:  8cn  8964  9re  8965  9pos  8982  6lt8  9069  5lt8  9070  4lt8  9071  3lt8  9072  2lt8  9073  1lt8  9074  8lt9  9075  7lt9  9076  8th4div3  9097  8lt10  9474  7lt10  9475  ef01bndlem  11719  cos2bnd  11723