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Theorem ssxr 11328
Description: The three (non-exclusive) possibilities implied by a subset of extended reals. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
ssxr (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))

Proof of Theorem ssxr
StepHypRef Expression
1 df-pr 4634 . . . . . . 7 {+∞, -∞} = ({+∞} ∪ {-∞})
21ineq2i 4225 . . . . . 6 (𝐴 ∩ {+∞, -∞}) = (𝐴 ∩ ({+∞} ∪ {-∞}))
3 indi 4290 . . . . . 6 (𝐴 ∩ ({+∞} ∪ {-∞})) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
42, 3eqtri 2763 . . . . 5 (𝐴 ∩ {+∞, -∞}) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
5 disjsn 4716 . . . . . . . 8 ((𝐴 ∩ {+∞}) = ∅ ↔ ¬ +∞ ∈ 𝐴)
6 disjsn 4716 . . . . . . . 8 ((𝐴 ∩ {-∞}) = ∅ ↔ ¬ -∞ ∈ 𝐴)
75, 6anbi12i 628 . . . . . . 7 (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴))
87biimpri 228 . . . . . 6 ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅))
9 pm4.56 990 . . . . . 6 ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) ↔ ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
10 un00 4451 . . . . . 6 (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
118, 9, 103imtr3i 291 . . . . 5 (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
124, 11eqtrid 2787 . . . 4 (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → (𝐴 ∩ {+∞, -∞}) = ∅)
13 reldisj 4459 . . . . 5 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})))
14 renfdisj 11319 . . . . . . . 8 (ℝ ∩ {+∞, -∞}) = ∅
15 disj3 4460 . . . . . . . 8 ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ℝ = (ℝ ∖ {+∞, -∞}))
1614, 15mpbi 230 . . . . . . 7 ℝ = (ℝ ∖ {+∞, -∞})
17 difun2 4487 . . . . . . 7 ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}) = (ℝ ∖ {+∞, -∞})
1816, 17eqtr4i 2766 . . . . . 6 ℝ = ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})
1918sseq2i 4025 . . . . 5 (𝐴 ⊆ ℝ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}))
2013, 19bitr4di 289 . . . 4 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ℝ))
2112, 20imbitrid 244 . . 3 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ))
2221orrd 863 . 2 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
23 df-xr 11297 . . 3 * = (ℝ ∪ {+∞, -∞})
2423sseq2i 4025 . 2 (𝐴 ⊆ ℝ*𝐴 ⊆ (ℝ ∪ {+∞, -∞}))
25 3orrot 1091 . . 3 ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴𝐴 ⊆ ℝ))
26 df-3or 1087 . . 3 ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴𝐴 ⊆ ℝ) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
2725, 26bitri 275 . 2 ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
2822, 24, 273imtr4i 292 1 (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3o 1085   = wceq 1537  wcel 2106  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631  {cpr 4633  cr 11152  +∞cpnf 11290  -∞cmnf 11291  *cxr 11292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297
This theorem is referenced by:  xrsupss  13348  xrinfmss  13349  xrssre  45298
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