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Theorem ssxr 10067
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 4158 . . . . . . 7 {+∞, -∞} = ({+∞} ∪ {-∞})
21ineq2i 3795 . . . . . 6 (𝐴 ∩ {+∞, -∞}) = (𝐴 ∩ ({+∞} ∪ {-∞}))
3 indi 3855 . . . . . 6 (𝐴 ∩ ({+∞} ∪ {-∞})) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
42, 3eqtri 2643 . . . . 5 (𝐴 ∩ {+∞, -∞}) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
5 disjsn 4223 . . . . . . . 8 ((𝐴 ∩ {+∞}) = ∅ ↔ ¬ +∞ ∈ 𝐴)
6 disjsn 4223 . . . . . . . 8 ((𝐴 ∩ {-∞}) = ∅ ↔ ¬ -∞ ∈ 𝐴)
75, 6anbi12i 732 . . . . . . 7 (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴))
87biimpri 218 . . . . . 6 ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅))
9 pm4.56 516 . . . . . 6 ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) ↔ ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
10 un00 3989 . . . . . 6 (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
118, 9, 103imtr3i 280 . . . . 5 (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
124, 11syl5eq 2667 . . . 4 (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → (𝐴 ∩ {+∞, -∞}) = ∅)
13 reldisj 3998 . . . . 5 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})))
14 renfdisj 10058 . . . . . . . 8 (ℝ ∩ {+∞, -∞}) = ∅
15 disj3 3999 . . . . . . . 8 ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ℝ = (ℝ ∖ {+∞, -∞}))
1614, 15mpbi 220 . . . . . . 7 ℝ = (ℝ ∖ {+∞, -∞})
17 difun2 4026 . . . . . . 7 ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}) = (ℝ ∖ {+∞, -∞})
1816, 17eqtr4i 2646 . . . . . 6 ℝ = ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})
1918sseq2i 3615 . . . . 5 (𝐴 ⊆ ℝ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}))
2013, 19syl6bbr 278 . . . 4 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ℝ))
2112, 20syl5ib 234 . . 3 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ))
2221orrd 393 . 2 (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
23 df-xr 10038 . . 3 * = (ℝ ∪ {+∞, -∞})
2423sseq2i 3615 . 2 (𝐴 ⊆ ℝ*𝐴 ⊆ (ℝ ∪ {+∞, -∞}))
25 3orrot 1042 . . 3 ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴𝐴 ⊆ ℝ))
26 df-3or 1037 . . 3 ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴𝐴 ⊆ ℝ) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
2725, 26bitri 264 . 2 ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
2822, 24, 273imtr4i 281 1 (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384  w3o 1035   = wceq 1480  wcel 1987  cdif 3557  cun 3558  cin 3559  wss 3560  c0 3897  {csn 4155  {cpr 4157  cr 9895  +∞cpnf 10031  -∞cmnf 10032  *cxr 10033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-resscn 9953
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038
This theorem is referenced by:  xrsupss  12098  xrinfmss  12099  xrssre  39063
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