Theorem xrltletr 10140

Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006.)

Assertion

Ref	Expression
xrltletr	|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B <_ C ) -> A < C ) )

Proof of Theorem xrltletr

Step	Hyp	Ref	Expression
1		simprr 533	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> B <_ C )
2		simpl2 1028	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> B e. RR* )
3		simpl3 1029	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> C e. RR* )
4		xrlenlt 8338	. . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B <_ C <-> -. C < B ) )
5	2, 3, 4	syl2anc 411	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> ( B <_ C <-> -. C < B ) )
6	1, 5	mpbid 147	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> -. C < B )
7		simprl 531	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> A < B )
8		xrltso 10129	. . . . . 6 |- < Or RR*
9		sowlin 4441	. . . . . 6 |- ( ( < Or RR* /\ ( A e. RR* /\ B e. RR* /\ C e. RR* ) ) -> ( A < B -> ( A < C \/ C < B ) ) )
10	8, 9	mpan 424	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> ( A < C \/ C < B ) ) )
11	10	adantr 276	. . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> ( A < B -> ( A < C \/ C < B ) ) )
12	7, 11	mpd 13	. . 3 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> ( A < C \/ C < B ) )
13	6, 12	ecased 1386	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A < B /\ B <_ C ) ) -> A < C )
14	13	ex 115	1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B <_ C ) -> A < C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   e. wcel 2203   class class class wbr 4109   Or wor 4416   RR*cxr 8307   < clt 8308   <_ cle 8309

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-po 4417  df-iso 4418  df-xp 4755  df-cnv 4757  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314

This theorem is referenced by:  xrltletrd  10144  xrre2  10154  xrre3  10155  ge0gtmnf  10156  iooss2  10250  iccssioo  10275  icossico  10276  icossioo  10297  ioossioo  10298  ioc0  10622