Theorem xrre3 10018

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 9979	. . . . . 6 |- ( B e. RR -> -oo < B )
2	1	adantl 277	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> -oo < B )
3		mnfxr 8203	. . . . . . 7 |- -oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> -oo e. RR* )
5		rexr 8192	. . . . . . 7 |- ( B e. RR -> B e. RR* )
6	5	adantl 277	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> B e. RR* )
7		simpl 109	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> A e. RR* )
8		xrltletr 10003	. . . . . 6 |- ( ( -oo e. RR* /\ B e. RR* /\ A e. RR* ) -> ( ( -oo < B /\ B <_ A ) -> -oo < A ) )
9	4, 6, 7, 8	syl3anc 1271	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( -oo < B /\ B <_ A ) -> -oo < A ) )
10	2, 9	mpand 429	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( B <_ A -> -oo < A ) )
11	10	imp 124	. . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ B <_ A ) -> -oo < A )
12	11	adantrr 479	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> -oo < A )
13		simprr 531	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A < +oo )
14		xrrebnd 10015	. . 3 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
15	14	ad2antrr 488	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
16	12, 13, 15	mpbir2and 950	1 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A e. RR )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   class class class wbr 4083   RRcr 7998   +oocpnf 8178   -oocmnf 8179   RR*cxr 8180   < 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-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113

This theorem depends on definitions:  df-bi 117  df-3or 1003  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-po 4387  df-iso 4388  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:  elicore  10486