Theorem lelttrdi 8055

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 961	. . . 4 |- ( ph -> A e. RR )
3	2	adantr 272	. . 3 |- ( ( ph /\ A < B ) -> A e. RR )
4	1	simp2d 962	. . . 4 |- ( ph -> B e. RR )
5	4	adantr 272	. . 3 |- ( ( ph /\ A < B ) -> B e. RR )
6	1	simp3d 963	. . . 4 |- ( ph -> C e. RR )
7	6	adantr 272	. . 3 |- ( ( ph /\ A < B ) -> C e. RR )
8		simpr 109	. . 3 |- ( ( ph /\ A < B ) -> A < B )
9		lelttrdi.l	. . . 4 |- ( ph -> B <_ C )
10	9	adantr 272	. . 3 |- ( ( ph /\ A < B ) -> B <_ C )
11	3, 5, 7, 8, 10	ltletrd 8052	. 2 |- ( ( ph /\ A < B ) -> A < C )
12	11	ex 114	1 |- ( ph -> ( A < B -> A < C ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 930   e. wcel 1448   class class class wbr 3875   RRcr 7499   < clt 7672   <_ cle 7673

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-pre-ltwlin 7608

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678

This theorem is referenced by:  subfzo0  9860