Theorem xnegmnf 10125

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 10068	. 2 |- -e -oo = if ( -oo = +oo , -oo , if ( -oo = -oo , +oo , -u -oo ) )
2		mnfnepnf 8294	. . 3 |- -oo =/= +oo
3		ifnefalse 3620	. . 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 3615	. 2 |- if ( -oo = -oo , +oo , -u -oo ) = +oo
7	1, 4, 6	3eqtri 2256	1 |- -e -oo = +oo

Syntax hints:   = wceq 1398   =/= wne 2403   ifcif 3607   +oocpnf 8270   -oocmnf 8271   -ucneg 8410   -ecxne 10065

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-un 4536  ax-cnex 8183

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-rex 2517  df-rab 2520  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-pnf 8275  df-mnf 8276  df-xr 8277  df-xneg 10068

This theorem is referenced by:  xnegcl  10128  xnegneg  10129  xltnegi  10131  xnegid  10155  xnegdi  10164  xsubge0  10177  xposdif  10178