Theorem ssxr 11313

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 4632	. . . . . . 7 ⊢ {+∞, -∞} = ({+∞} ∪ {-∞})
2	1	ineq2i 4208	. . . . . 6 ⊢ (𝐴 ∩ {+∞, -∞}) = (𝐴 ∩ ({+∞} ∪ {-∞}))
3		indi 4273	. . . . . 6 ⊢ (𝐴 ∩ ({+∞} ∪ {-∞})) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
4	2, 3	eqtri 2753	. . . . 5 ⊢ (𝐴 ∩ {+∞, -∞}) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
5		disjsn 4716	. . . . . . . 8 ⊢ ((𝐴 ∩ {+∞}) = ∅ ↔ ¬ +∞ ∈ 𝐴)
6		disjsn 4716	. . . . . . . 8 ⊢ ((𝐴 ∩ {-∞}) = ∅ ↔ ¬ -∞ ∈ 𝐴)
7	5, 6	anbi12i 626	. . . . . . 7 ⊢ (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴))
8	7	biimpri 227	. . . . . 6 ⊢ ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅))
9		pm4.56 986	. . . . . 6 ⊢ ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) ↔ ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
10		un00 4443	. . . . . 6 ⊢ (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
11	8, 9, 10	3imtr3i 290	. . . . 5 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
12	4, 11	eqtrid 2777	. . . 4 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → (𝐴 ∩ {+∞, -∞}) = ∅)
13		reldisj 4452	. . . . 5 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})))
14		renfdisj 11304	. . . . . . . 8 ⊢ (ℝ ∩ {+∞, -∞}) = ∅
15		disj3 4454	. . . . . . . 8 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ℝ = (ℝ ∖ {+∞, -∞}))
16	14, 15	mpbi 229	. . . . . . 7 ⊢ ℝ = (ℝ ∖ {+∞, -∞})
17		difun2 4481	. . . . . . 7 ⊢ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}) = (ℝ ∖ {+∞, -∞})
18	16, 17	eqtr4i 2756	. . . . . 6 ⊢ ℝ = ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})
19	18	sseq2i 4007	. . . . 5 ⊢ (𝐴 ⊆ ℝ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}))
20	13, 19	bitr4di 288	. . . 4 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ℝ))
21	12, 20	imbitrid 243	. . 3 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ))
22	21	orrd 861	. 2 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
23		df-xr 11282	. . 3 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
24	23	sseq2i 4007	. 2 ⊢ (𝐴 ⊆ ℝ* ↔ 𝐴 ⊆ (ℝ ∪ {+∞, -∞}))
25		3orrot 1089	. . 3 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴 ∨ 𝐴 ⊆ ℝ))
26		df-3or 1085	. . 3 ⊢ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴 ∨ 𝐴 ⊆ ℝ) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
27	25, 26	bitri 274	. 2 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
28	22, 24, 27	3imtr4i 291	1 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ wo 845   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098   ∖ cdif 3942   ∪ cun 3943   ∩ cin 3944   ⊆ wss 3945  ∅c0 4323  {csn 4629  {cpr 4631  ℝcr 11137  +∞cpnf 11275  -∞cmnf 11276  ℝ^*cxr 11277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-resscn 11195

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282

This theorem is referenced by:  xrsupss  13320  xrinfmss  13321  xrssre  44793