Theorem ssxr 11267

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

Step	Hyp	Ref	Expression
1		df-pr 4588	. . . . . . 7 ⊢ {+∞, -∞} = ({+∞} ∪ {-∞})
2	1	ineq2i 4172	. . . . . 6 ⊢ (𝐴 ∩ {+∞, -∞}) = (𝐴 ∩ ({+∞} ∪ {-∞}))
3		indi 4239	. . . . . 6 ⊢ (𝐴 ∩ ({+∞} ∪ {-∞})) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
4	2, 3	eqtri 2788	. . . . 5 ⊢ (𝐴 ∩ {+∞, -∞}) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
5		disjsn 4673	. . . . . . . 8 ⊢ ((𝐴 ∩ {+∞}) = ∅ ↔ ¬ +∞ ∈ 𝐴)
6		disjsn 4673	. . . . . . . 8 ⊢ ((𝐴 ∩ {-∞}) = ∅ ↔ ¬ -∞ ∈ 𝐴)
7	5, 6	anbi12i 639	. . . . . . 7 ⊢ (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴))
8	7	biimpri 231	. . . . . 6 ⊢ ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅))
9		pm4.56 1004	. . . . . 6 ⊢ ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) ↔ ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
10		un00 4402	. . . . . 6 ⊢ (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
11	8, 9, 10	3imtr3i 294	. . . . 5 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
12	4, 11	eqtrid 2812	. . . 4 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → (𝐴 ∩ {+∞, -∞}) = ∅)
13		reldisj 4410	. . . . 5 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})))
14		renfdisj 11257	. . . . . . . 8 ⊢ (ℝ ∩ {+∞, -∞}) = ∅
15		disj3 4411	. . . . . . . 8 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ℝ = (ℝ ∖ {+∞, -∞}))
16	14, 15	mpbi 233	. . . . . . 7 ⊢ ℝ = (ℝ ∖ {+∞, -∞})
17		difun2 4438	. . . . . . 7 ⊢ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}) = (ℝ ∖ {+∞, -∞})
18	16, 17	eqtr4i 2791	. . . . . 6 ⊢ ℝ = ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})
19	18	sseq2i 3968	. . . . 5 ⊢ (𝐴 ⊆ ℝ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}))
20	13, 19	bitr4di 292	. . . 4 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ℝ))
21	12, 20	imbitrid 247	. . 3 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ))
22	21	orrd 876	. 2 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
23		df-xr 11235	. . 3 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
24	23	sseq2i 3968	. 2 ⊢ (𝐴 ⊆ ℝ* ↔ 𝐴 ⊆ (ℝ ∪ {+∞, -∞}))
25		3orrot 1106	. . 3 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴 ∨ 𝐴 ⊆ ℝ))
26		df-3or 1102	. . 3 ⊢ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴 ∨ 𝐴 ⊆ ℝ) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
27	25, 26	bitri 278	. 2 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
28	22, 24, 27	3imtr4i 295	1 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   = wceq 1563   ∈ wcel 2145   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  {csn 4585  {cpr 4587  ℝcr 11087  +∞cpnf 11228  -∞cmnf 11229  ℝ^*cxr 11230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-resscn 11145

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235

This theorem is referenced by:  xrsupss  13326  xrinfmss  13327  xrssre  45922