Theorem lelttrdi 8345

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 1004	. . . 4 |- ( ph -> A e. RR )
3	2	adantr 274	. . 3 |- ( ( ph /\ A < B ) -> A e. RR )
4	1	simp2d 1005	. . . 4 |- ( ph -> B e. RR )
5	4	adantr 274	. . 3 |- ( ( ph /\ A < B ) -> B e. RR )
6	1	simp3d 1006	. . . 4 |- ( ph -> C e. RR )
7	6	adantr 274	. . 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 274	. . 3 |- ( ( ph /\ A < B ) -> B <_ C )
11	3, 5, 7, 8, 10	ltletrd 8342	. 2 |- ( ( ph /\ A < B ) -> A < C )
12	11	ex 114	1 |- ( ph -> ( A < B -> A < C ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   e. wcel 2141   class class class wbr 3989   RRcr 7773   < clt 7954   <_ cle 7955

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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltwlin 7887

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960

This theorem is referenced by:  difgtsumgt  9281  subfzo0  10198