Theorem xrre3 10158

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 10119	. . . . . 6 |- ( B e. RR -> -oo < B )
2	1	adantl 277	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> -oo < B )
3		mnfxr 8332	. . . . . . 7 |- -oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> -oo e. RR* )
5		rexr 8321	. . . . . . 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 10143	. . . . . 6 |- ( ( -oo e. RR* /\ B e. RR* /\ A e. RR* ) -> ( ( -oo < B /\ B <_ A ) -> -oo < A ) )
9	4, 6, 7, 8	syl3anc 1274	. . . . 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 533	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A < +oo )
14		xrrebnd 10155	. . 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 953	1 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A e. RR )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2205   class class class wbr 4111   RRcr 8128   +oocpnf 8307   -oocmnf 8308   RR*cxr 8309   < clt 8310   <_ cle 8311

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-po 4419  df-iso 4420  df-xp 4757  df-cnv 4759  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316

This theorem is referenced by:  elicore  10630