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Theorem ssxr 10703
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 4531 . . . . . . 7 {+∞, -∞} = ({+∞} ∪ {-∞})
21ineq2i 4139 . . . . . 6 (𝐴 ∩ {+∞, -∞}) = (𝐴 ∩ ({+∞} ∪ {-∞}))
3 indi 4203 . . . . . 6 (𝐴 ∩ ({+∞} ∪ {-∞})) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
42, 3eqtri 2824 . . . . 5 (𝐴 ∩ {+∞, -∞}) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
5 disjsn 4610 . . . . . . . 8 ((𝐴 ∩ {+∞}) = ∅ ↔ ¬ +∞ ∈ 𝐴)
6 disjsn 4610 . . . . . . . 8 ((𝐴 ∩ {-∞}) = ∅ ↔ ¬ -∞ ∈ 𝐴)
75, 6anbi12i 629 . . . . . . 7 (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴))
87biimpri 231 . . . . . 6 ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅))
9 pm4.56 986 . . . . . 6 ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) ↔ ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
10 un00 4353 . . . . . 6 (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
118, 9, 103imtr3i 294 . . . . 5 (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
124, 11syl5eq 2848 . . . 4 (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → (𝐴 ∩ {+∞, -∞}) = ∅)
13 reldisj 4362 . . . . 5 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})))
14 renfdisj 10694 . . . . . . . 8 (ℝ ∩ {+∞, -∞}) = ∅
15 disj3 4364 . . . . . . . 8 ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ℝ = (ℝ ∖ {+∞, -∞}))
1614, 15mpbi 233 . . . . . . 7 ℝ = (ℝ ∖ {+∞, -∞})
17 difun2 4390 . . . . . . 7 ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}) = (ℝ ∖ {+∞, -∞})
1816, 17eqtr4i 2827 . . . . . 6 ℝ = ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})
1918sseq2i 3947 . . . . 5 (𝐴 ⊆ ℝ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}))
2013, 19syl6bbr 292 . . . 4 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ℝ))
2112, 20syl5ib 247 . . 3 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ))
2221orrd 860 . 2 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
23 df-xr 10672 . . 3 * = (ℝ ∪ {+∞, -∞})
2423sseq2i 3947 . 2 (𝐴 ⊆ ℝ*𝐴 ⊆ (ℝ ∪ {+∞, -∞}))
25 3orrot 1089 . . 3 ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴𝐴 ⊆ ℝ))
26 df-3or 1085 . . 3 ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴𝐴 ⊆ ℝ) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
2725, 26bitri 278 . 2 ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
2822, 24, 273imtr4i 295 1 (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  w3o 1083   = wceq 1538  wcel 2112  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  {csn 4528  {cpr 4530  cr 10529  +∞cpnf 10665  -∞cmnf 10666  *cxr 10667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-resscn 10587
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672
This theorem is referenced by:  xrsupss  12694  xrinfmss  12695  xrssre  41967
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