Theorem xnegmnf 9786

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 9729	. 2 |- -e -oo = if ( -oo = +oo , -oo , if ( -oo = -oo , +oo , -u -oo ) )
2		mnfnepnf 7975	. . 3 |- -oo =/= +oo
3		ifnefalse 3537	. . 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 2170	. . 3 |- -oo = -oo
6	5	iftruei 3532	. 2 |- if ( -oo = -oo , +oo , -u -oo ) = +oo
7	1, 4, 6	3eqtri 2195	1 |- -e -oo = +oo

Syntax hints:   = wceq 1348   =/= wne 2340   ifcif 3526   +oocpnf 7951   -oocmnf 7952   -ucneg 8091   -ecxne 9726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-un 4418  ax-cnex 7865

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-pnf 7956  df-mnf 7957  df-xr 7958  df-xneg 9729

This theorem is referenced by:  xnegcl  9789  xnegneg  9790  xltnegi  9792  xnegid  9816  xnegdi  9825  xsubge0  9838  xposdif  9839