Theorem ssxr 11309

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 4609	. . . . . . 7 ⊢ {+∞, -∞} = ({+∞} ∪ {-∞})
2	1	ineq2i 4197	. . . . . 6 ⊢ (𝐴 ∩ {+∞, -∞}) = (𝐴 ∩ ({+∞} ∪ {-∞}))
3		indi 4264	. . . . . 6 ⊢ (𝐴 ∩ ({+∞} ∪ {-∞})) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
4	2, 3	eqtri 2759	. . . . 5 ⊢ (𝐴 ∩ {+∞, -∞}) = ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞}))
5		disjsn 4692	. . . . . . . 8 ⊢ ((𝐴 ∩ {+∞}) = ∅ ↔ ¬ +∞ ∈ 𝐴)
6		disjsn 4692	. . . . . . . 8 ⊢ ((𝐴 ∩ {-∞}) = ∅ ↔ ¬ -∞ ∈ 𝐴)
7	5, 6	anbi12i 628	. . . . . . 7 ⊢ (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ (¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴))
8	7	biimpri 228	. . . . . 6 ⊢ ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅))
9		pm4.56 990	. . . . . 6 ⊢ ((¬ +∞ ∈ 𝐴 ∧ ¬ -∞ ∈ 𝐴) ↔ ¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))
10		un00 4425	. . . . . 6 ⊢ (((𝐴 ∩ {+∞}) = ∅ ∧ (𝐴 ∩ {-∞}) = ∅) ↔ ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
11	8, 9, 10	3imtr3i 291	. . . . 5 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → ((𝐴 ∩ {+∞}) ∪ (𝐴 ∩ {-∞})) = ∅)
12	4, 11	eqtrid 2783	. . . 4 ⊢ (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → (𝐴 ∩ {+∞, -∞}) = ∅)
13		reldisj 4433	. . . . 5 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})))
14		renfdisj 11300	. . . . . . . 8 ⊢ (ℝ ∩ {+∞, -∞}) = ∅
15		disj3 4434	. . . . . . . 8 ⊢ ((ℝ ∩ {+∞, -∞}) = ∅ ↔ ℝ = (ℝ ∖ {+∞, -∞}))
16	14, 15	mpbi 230	. . . . . . 7 ⊢ ℝ = (ℝ ∖ {+∞, -∞})
17		difun2 4461	. . . . . . 7 ⊢ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}) = (ℝ ∖ {+∞, -∞})
18	16, 17	eqtr4i 2762	. . . . . 6 ⊢ ℝ = ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞})
19	18	sseq2i 3993	. . . . 5 ⊢ (𝐴 ⊆ ℝ ↔ 𝐴 ⊆ ((ℝ ∪ {+∞, -∞}) ∖ {+∞, -∞}))
20	13, 19	bitr4di 289	. . . 4 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((𝐴 ∩ {+∞, -∞}) = ∅ ↔ 𝐴 ⊆ ℝ))
21	12, 20	imbitrid 244	. . 3 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → (¬ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) → 𝐴 ⊆ ℝ))
22	21	orrd 863	. 2 ⊢ (𝐴 ⊆ (ℝ ∪ {+∞, -∞}) → ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
23		df-xr 11278	. . 3 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
24	23	sseq2i 3993	. 2 ⊢ (𝐴 ⊆ ℝ* ↔ 𝐴 ⊆ (ℝ ∪ {+∞, -∞}))
25		3orrot 1091	. . 3 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ (+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴 ∨ 𝐴 ⊆ ℝ))
26		df-3or 1087	. . 3 ⊢ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴 ∨ 𝐴 ⊆ ℝ) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
27	25, 26	bitri 275	. 2 ⊢ ((𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ↔ ((+∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴) ∨ 𝐴 ⊆ ℝ))
28	22, 24, 27	3imtr4i 292	1 ⊢ (𝐴 ⊆ ℝ* → (𝐴 ⊆ ℝ ∨ +∞ ∈ 𝐴 ∨ -∞ ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109   ∖ cdif 3928   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606  {cpr 4608  ℝcr 11133  +∞cpnf 11271  -∞cmnf 11272  ℝ^*cxr 11273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-resscn 11191

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278

This theorem is referenced by:  xrsupss  13330  xrinfmss  13331  xrssre  45342