Theorem xnegpnf 10024

Description: Minus +oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)

Assertion

Ref	Expression
xnegpnf	|- -e +oo = -oo

Proof of Theorem xnegpnf

Step	Hyp	Ref	Expression
1		df-xneg 9968	. 2 |- -e +oo = if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) )
2		eqid 2229	. . 3 |- +oo = +oo
3	2	iftruei 3608	. 2 |- if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) ) = -oo
4	1, 3	eqtri 2250	1 |- -e +oo = -oo

Syntax hints:   = wceq 1395   ifcif 3602   +oocpnf 8178   -oocmnf 8179   -ucneg 8318   -ecxne 9965

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-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3603  df-xneg 9968

This theorem is referenced by:  xnegcl  10028  xnegneg  10029  xltnegi  10031  xnegid  10055  xnegdi  10064  xaddass2  10066  xsubge0  10077  xposdif  10078  xlesubadd  10079  xblss2ps  15078  xblss2  15079