Theorem mnfltxr 9852

Description: Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006.)

Assertion

Ref	Expression
mnfltxr	|- ( ( A e. RR \/ A = +oo ) -> -oo < A )

Proof of Theorem mnfltxr

Step	Hyp	Ref	Expression
1		mnflt 9849	. 2 |- ( A e. RR -> -oo < A )
2		mnfltpnf 9851	. . 3 |- -oo < +oo
3		breq2 4033	. . 3 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
4	2, 3	mpbiri 168	. 2 |- ( A = +oo -> -oo < A )
5	1, 4	jaoi 717	1 |- ( ( A e. RR \/ A = +oo ) -> -oo < A )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2164   class class class wbr 4029   RRcr 7871   +oocpnf 8051   -oocmnf 8052   < clt 8054

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059

This theorem is referenced by:  xrltso  9862