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Theorem elxr 8414
Description: Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
elxr (A * ↔ (A A = +∞ A = -∞))

Proof of Theorem elxr
StepHypRef Expression
1 df-xr 6813 . . 3 * = (ℝ ∪ {+∞, -∞})
21eleq2i 2101 . 2 (A *A (ℝ ∪ {+∞, -∞}))
3 elun 3078 . 2 (A (ℝ ∪ {+∞, -∞}) ↔ (A A {+∞, -∞}))
4 pnfex 8411 . . . . 5 +∞ V
5 mnfxr 8412 . . . . . 6 -∞ *
65elexi 2561 . . . . 5 -∞ V
74, 6elpr2 3385 . . . 4 (A {+∞, -∞} ↔ (A = +∞ A = -∞))
87orbi2i 678 . . 3 ((A A {+∞, -∞}) ↔ (A (A = +∞ A = -∞)))
9 3orass 887 . . 3 ((A A = +∞ A = -∞) ↔ (A (A = +∞ A = -∞)))
108, 9bitr4i 176 . 2 ((A A {+∞, -∞}) ↔ (A A = +∞ A = -∞))
112, 3, 103bitri 195 1 (A * ↔ (A A = +∞ A = -∞))
Colors of variables: wff set class
Syntax hints:  wb 98   wo 628   w3o 883   = wceq 1242   wcel 1390  cun 2909  {cpr 3367  cr 6662  +∞cpnf 6806  -∞cmnf 6807  *cxr 6808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-un 4135  ax-cnex 6726
This theorem depends on definitions:  df-bi 110  df-3or 885  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-uni 3571  df-pnf 6811  df-mnf 6812  df-xr 6813
This theorem is referenced by:  xrnemnf  8417  xrnepnf  8418  xrltnr  8419  xrltnsym  8432  xrlttr  8434  xrltso  8435  xrlttri3  8436  nltpnft  8448  ngtmnft  8449  xrrebnd  8450  xnegcl  8463  xnegneg  8464  xltnegi  8466
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