Theorem lelttrdi 8573

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 1033	. . . 4 |- ( ph -> A e. RR )
3	2	adantr 276	. . 3 |- ( ( ph /\ A < B ) -> A e. RR )
4	1	simp2d 1034	. . . 4 |- ( ph -> B e. RR )
5	4	adantr 276	. . 3 |- ( ( ph /\ A < B ) -> B e. RR )
6	1	simp3d 1035	. . . 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 8570	. 2 |- ( ( ph /\ A < B ) -> A < C )
12	11	ex 115	1 |- ( ph -> ( A < B -> A < C ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   e. wcel 2200   class class class wbr 4083   RRcr 7998   < clt 8181   <_ cle 8182

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-pre-ltwlin 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187

This theorem is referenced by:  difgtsumgt  9516  subfzo0  10448