Theorem xnegpnf 9860

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 9804	. 2 |- -e +oo = if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) )
2		eqid 2189	. . 3 |- +oo = +oo
3	2	iftruei 3555	. 2 |- if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) ) = -oo
4	1, 3	eqtri 2210	1 |- -e +oo = -oo

Syntax hints:   = wceq 1364   ifcif 3549   +oocpnf 8020   -oocmnf 8021   -ucneg 8160   -ecxne 9801

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 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-if 3550  df-xneg 9804

This theorem is referenced by:  xnegcl  9864  xnegneg  9865  xltnegi  9867  xnegid  9891  xnegdi  9900  xaddass2  9902  xsubge0  9913  xposdif  9914  xlesubadd  9915  xblss2ps  14381  xblss2  14382