Theorem xnegmnf 10063

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 10006	. 2 |- -e -oo = if ( -oo = +oo , -oo , if ( -oo = -oo , +oo , -u -oo ) )
2		mnfnepnf 8234	. . 3 |- -oo =/= +oo
3		ifnefalse 3616	. . 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 2231	. . 3 |- -oo = -oo
6	5	iftruei 3611	. 2 |- if ( -oo = -oo , +oo , -u -oo ) = +oo
7	1, 4, 6	3eqtri 2256	1 |- -e -oo = +oo

Syntax hints:   = wceq 1397   =/= wne 2402   ifcif 3605   +oocpnf 8210   -oocmnf 8211   -ucneg 8350   -ecxne 10003

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 619  ax-in2 620  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-un 4530  ax-cnex 8122

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-uni 3894  df-pnf 8215  df-mnf 8216  df-xr 8217  df-xneg 10006

This theorem is referenced by:  xnegcl  10066  xnegneg  10067  xltnegi  10069  xnegid  10093  xnegdi  10102  xsubge0  10115  xposdif  10116