Theorem lelttrdi 8445

Description: If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)

Hypotheses

Ref	Expression
lelttrdi.r	|- ( ph -> ( A e. RR /\ B e. RR /\ C e. RR ) )
lelttrdi.l	|- ( ph -> B <_ C )

Assertion

Ref	Expression
lelttrdi	|- ( ph -> ( A < B -> A < C ) )

Proof of Theorem lelttrdi

Step	Hyp	Ref	Expression
1		lelttrdi.r	. . . . 5 |- ( ph -> ( A e. RR /\ B e. RR /\ C e. RR ) )
2	1	simp1d 1011	. . . 4 |- ( ph -> A e. RR )
3	2	adantr 276	. . 3 |- ( ( ph /\ A < B ) -> A e. RR )
4	1	simp2d 1012	. . . 4 |- ( ph -> B e. RR )
5	4	adantr 276	. . 3 |- ( ( ph /\ A < B ) -> B e. RR )
6	1	simp3d 1013	. . . 4 |- ( ph -> C e. RR )
7	6	adantr 276	. . 3 |- ( ( ph /\ A < B ) -> C e. RR )
8		simpr 110	. . 3 |- ( ( ph /\ A < B ) -> A < B )
9		lelttrdi.l	. . . 4 |- ( ph -> B <_ C )
10	9	adantr 276	. . 3 |- ( ( ph /\ A < B ) -> B <_ C )
11	3, 5, 7, 8, 10	ltletrd 8442	. 2 |- ( ( ph /\ A < B ) -> A < C )
12	11	ex 115	1 |- ( ph -> ( A < B -> A < C ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   e. wcel 2164   class class class wbr 4029   RRcr 7871   < clt 8054   <_ cle 8055

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-pre-ltwlin 7985

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060

This theorem is referenced by:  difgtsumgt  9386  subfzo0  10309