Theorem xrre3 9216

Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)

Assertion

Ref	Expression
xrre3	|- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A e. RR )

Proof of Theorem xrre3

Step	Hyp	Ref	Expression
1		mnflt 9185	. . . . . 6 |- ( B e. RR -> -oo < B )
2	1	adantl 271	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> -oo < B )
3		mnfxr 7488	. . . . . . 7 |- -oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> -oo e. RR* )
5		rexr 7477	. . . . . . 7 |- ( B e. RR -> B e. RR* )
6	5	adantl 271	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> B e. RR* )
7		simpl 107	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> A e. RR* )
8		xrltletr 9204	. . . . . 6 |- ( ( -oo e. RR* /\ B e. RR* /\ A e. RR* ) -> ( ( -oo < B /\ B <_ A ) -> -oo < A ) )
9	4, 6, 7, 8	syl3anc 1172	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( -oo < B /\ B <_ A ) -> -oo < A ) )
10	2, 9	mpand 420	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( B <_ A -> -oo < A ) )
11	10	imp 122	. . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ B <_ A ) -> -oo < A )
12	11	adantrr 463	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> -oo < A )
13		simprr 499	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A < +oo )
14		xrrebnd 9213	. . 3 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
15	14	ad2antrr 472	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
16	12, 13, 15	mpbir2and 888	1 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A e. RR )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1436   class class class wbr 3820   RRcr 7293   +oocpnf 7463   -oocmnf 7464   RR*cxr 7465   < clt 7466   <_ cle 7467

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403

This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-po 4097  df-iso 4098  df-xp 4417  df-cnv 4419  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472

This theorem is referenced by: (None)