Theorem xnegpnf 9952

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 9896	. 2 |- -e +oo = if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) )
2		eqid 2205	. . 3 |- +oo = +oo
3	2	iftruei 3577	. 2 |- if ( +oo = +oo , -oo , if ( +oo = -oo , +oo , -u +oo ) ) = -oo
4	1, 3	eqtri 2226	1 |- -e +oo = -oo

Syntax hints:   = wceq 1373   ifcif 3571   +oocpnf 8106   -oocmnf 8107   -ucneg 8246   -ecxne 9893

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-if 3572  df-xneg 9896

This theorem is referenced by:  xnegcl  9956  xnegneg  9957  xltnegi  9959  xnegid  9983  xnegdi  9992  xaddass2  9994  xsubge0  10005  xposdif  10006  xlesubadd  10007  xblss2ps  14909  xblss2  14910