Theorem xnegpnf 9950

Description: Minus +∞. Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)

Assertion

Ref	Expression
xnegpnf	⊢ -𝑒+∞ = -∞

Proof of Theorem xnegpnf

Step	Hyp	Ref	Expression
1		df-xneg 9894	. 2 ⊢ -𝑒+∞ = if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞))
2		eqid 2205	. . 3 ⊢ +∞ = +∞
3	2	iftruei 3577	. 2 ⊢ if(+∞ = +∞, -∞, if(+∞ = -∞, +∞, -+∞)) = -∞
4	1, 3	eqtri 2226	1 ⊢ -𝑒+∞ = -∞

Syntax hints:   = wceq 1373  ifcif 3571  +∞cpnf 8104  -∞cmnf 8105  -cneg 8244  -_𝑒cxne 9891

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-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-if 3572  df-xneg 9894

This theorem is referenced by:  xnegcl  9954  xnegneg  9955  xltnegi  9957  xnegid  9981  xnegdi  9990  xaddass2  9992  xsubge0  10003  xposdif  10004  xlesubadd  10005  xblss2ps  14876  xblss2  14877