Theorem mnfltxr 10119

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 10116	. 2 |- ( A e. RR -> -oo < A )
2		mnfltpnf 10118	. . 3 |- -oo < +oo
3		breq2 4113	. . 3 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
4	2, 3	mpbiri 168	. 2 |- ( A = +oo -> -oo < A )
5	1, 4	jaoi 724	1 |- ( ( A e. RR \/ A = +oo ) -> -oo < A )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2203   class class class wbr 4109   RRcr 8126   +oocpnf 8305   -oocmnf 8306   < clt 8308

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 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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313

This theorem is referenced by:  xrltso  10129