Theorem xnegmnf 9898

Description: Minus -oo. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)

Assertion

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

Proof of Theorem xnegmnf

Step	Hyp	Ref	Expression
1		df-xneg 9841	. 2 |- -e -oo = if ( -oo = +oo , -oo , if ( -oo = -oo , +oo , -u -oo ) )
2		mnfnepnf 8077	. . 3 |- -oo =/= +oo
3		ifnefalse 3569	. . 3 |- ( -oo =/= +oo -> if ( -oo = +oo , -oo , if ( -oo = -oo , +oo , -u -oo ) ) = if ( -oo = -oo , +oo , -u -oo ) )
4	2, 3	ax-mp 5	. 2 |- if ( -oo = +oo , -oo , if ( -oo = -oo , +oo , -u -oo ) ) = if ( -oo = -oo , +oo , -u -oo )
5		eqid 2193	. . 3 |- -oo = -oo
6	5	iftruei 3564	. 2 |- if ( -oo = -oo , +oo , -u -oo ) = +oo
7	1, 4, 6	3eqtri 2218	1 |- -e -oo = +oo

Syntax hints:   = wceq 1364   =/= wne 2364   ifcif 3558   +oocpnf 8053   -oocmnf 8054   -ucneg 8193   -ecxne 9838

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-in1 615  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-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-un 4465  ax-cnex 7965

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-rex 2478  df-rab 2481  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-pnf 8058  df-mnf 8059  df-xr 8060  df-xneg 9841

This theorem is referenced by:  xnegcl  9901  xnegneg  9902  xltnegi  9904  xnegid  9928  xnegdi  9937  xsubge0  9950  xposdif  9951