Theorem mnfltxr 8707

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 8704	. 2 |- ( A e. RR -> -oo < A )
2		mnfltpnf 8706	. . 3 |- -oo < +oo
3		breq2 3768	. . 3 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
4	2, 3	mpbiri 157	. 2 |- ( A = +oo -> -oo < A )
5	1, 4	jaoi 636	1 |- ( ( A e. RR \/ A = +oo ) -> -oo < A )

Syntax hints:   -> wi 4   \/ wo 629   = wceq 1243   e. wcel 1393   class class class wbr 3764   RRcr 6888   +oocpnf 7057   -oocmnf 7058   < clt 7060

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-cnex 6975

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065

This theorem is referenced by:  xrltso  8717