Theorem xrre3 10118

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 10079	. . . . . 6 |- ( B e. RR -> -oo < B )
2	1	adantl 277	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> -oo < B )
3		mnfxr 8295	. . . . . . 7 |- -oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> -oo e. RR* )
5		rexr 8284	. . . . . . 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 10103	. . . . . 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 10115	. . 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 2202   class class class wbr 4093   RRcr 8091   +oocpnf 8270   -oocmnf 8271   RR*cxr 8272   < clt 8273   <_ cle 8274

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-po 4399  df-iso 4400  df-xp 4737  df-cnv 4739  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279

This theorem is referenced by:  elicore  10589