Theorem xnegmnf 12041

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 11946	. 2 |- -e -oo = if ( -oo = +oo , -oo , if ( -oo = -oo , +oo , -u -oo ) )
2		mnfnepnf 10095	. . 3 |- -oo =/= +oo
3		ifnefalse 4098	. . 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 2622	. . 3 |- -oo = -oo
6	5	iftruei 4093	. 2 |- if ( -oo = -oo , +oo , -u -oo ) = +oo
7	1, 4, 6	3eqtri 2648	1 |- -e -oo = +oo

Syntax hints:   = wceq 1483   =/= wne 2794   ifcif 4086   +oocpnf 10071   -oocmnf 10072   -ucneg 10267   -ecxne 11943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-pow 4843  ax-un 6949  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-rex 2918  df-rab 2921  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-pnf 10076  df-mnf 10077  df-xr 10078  df-xneg 11946

This theorem is referenced by:  xnegcl  12044  xnegneg  12045  xltnegi  12047  xnegid  12069  xnegdi  12078  xsubge0  12091  xmulneg1  12099  xmulpnf1n  12108  xadddi2  12127  xrsdsreclblem  19792  xaddeq0  29518  xrge0npcan  29694  carsgclctunlem2  30381  supminfxr  39694  supminfxr2  39699  liminf0  40025