Theorem lelttrdi 8381

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 1009	. . . 4 |- ( ph -> A e. RR )
3	2	adantr 276	. . 3 |- ( ( ph /\ A < B ) -> A e. RR )
4	1	simp2d 1010	. . . 4 |- ( ph -> B e. RR )
5	4	adantr 276	. . 3 |- ( ( ph /\ A < B ) -> B e. RR )
6	1	simp3d 1011	. . . 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 8378	. 2 |- ( ( ph /\ A < B ) -> A < C )
12	11	ex 115	1 |- ( ph -> ( A < B -> A < C ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   e. wcel 2148   class class class wbr 4003   RRcr 7809   < clt 7990   <_ cle 7991

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltwlin 7923

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996

This theorem is referenced by:  difgtsumgt  9320  subfzo0  10239