Theorem mnfltxr 9671

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 9668	. 2 |- ( A e. RR -> -oo < A )
2		mnfltpnf 9670	. . 3 |- -oo < +oo
3		breq2 3965	. . 3 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
4	2, 3	mpbiri 167	. 2 |- ( A = +oo -> -oo < A )
5	1, 4	jaoi 706	1 |- ( ( A e. RR \/ A = +oo ) -> -oo < A )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1332   e. wcel 2125   class class class wbr 3961   RRcr 7710   +oocpnf 7888   -oocmnf 7889   < clt 7891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-cnex 7802

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-xp 4585  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896

This theorem is referenced by:  xrltso  9681