Theorem mnfltxr 10020

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 10017	. 2 |- ( A e. RR -> -oo < A )
2		mnfltpnf 10019	. . 3 |- -oo < +oo
3		breq2 4092	. . 3 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
4	2, 3	mpbiri 168	. 2 |- ( A = +oo -> -oo < A )
5	1, 4	jaoi 723	1 |- ( ( A e. RR \/ A = +oo ) -> -oo < A )

Syntax hints:   -> wi 4   \/ wo 715   = wceq 1397   e. wcel 2202   class class class wbr 4088   RRcr 8030   +oocpnf 8210   -oocmnf 8211   < clt 8213

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218

This theorem is referenced by:  xrltso  10030