Theorem ssxr 6714

Description: The three (non-exclusive) possibilities implied by a subset of extended reals. (The proof was shortened by Andrew Salmon, 19-Nov-2011.)

Assertion

Ref	Expression
ssxr	|- (A C_ RR* -> (A C_ RR \/ +oo e. A \/ -oo e. A))

Proof of Theorem ssxr

Step	Hyp	Ref	Expression
1		reldisj 2916	. . . . 5 |- (A C_ (RR u. { +oo, -oo}) -> ((A i^i { +oo, -oo}) = (/) <-> A C_ ((RR u. { +oo, -oo}) \ { +oo, -oo})))
2		renfdisj 6712	. . . . . . . 8 |- (RR i^i { +oo, -oo}) = (/)
3		disj3 2918	. . . . . . . 8 |- ((RR i^i { +oo, -oo}) = (/) <-> RR = (RR \ { +oo, -oo}))
4	2, 3	mpbi 206	. . . . . . 7 |- RR = (RR \ { +oo, -oo})
5		difun2 2953	. . . . . . 7 |- ((RR u. { +oo, -oo}) \ { +oo, -oo}) = (RR \ { +oo, -oo})
6	4, 5	eqtr4i 1911	. . . . . 6 |- RR = ((RR u. { +oo, -oo}) \ { +oo, -oo})
7	6	sseq2i 2642	. . . . 5 |- (A C_ RR <-> A C_ ((RR u. { +oo, -oo}) \ { +oo, -oo}))
8	1, 7	syl6bbr 597	. . . 4 |- (A C_ (RR u. { +oo, -oo}) -> ((A i^i { +oo, -oo}) = (/) <-> A C_ RR))
9		disjsn 3089	. . . . . . . 8 |- ((A i^i { +oo}) = (/) <-> -. +oo e. A)
10		disjsn 3089	. . . . . . . 8 |- ((A i^i { -oo}) = (/) <-> -. -oo e. A)
11	9, 10	anbi12i 540	. . . . . . 7 |- (((A i^i { +oo}) = (/) /\ (A i^i { -oo}) = (/)) <-> (-. +oo e. A /\ -. -oo e. A))
12	11	biimpri 169	. . . . . 6 |- ((-. +oo e. A /\ -. -oo e. A) -> ((A i^i { +oo}) = (/) /\ (A i^i { -oo}) = (/)))
13		pm4.56 337	. . . . . 6 |- ((-. +oo e. A /\ -. -oo e. A) <-> -. ( +oo e. A \/ -oo e. A))
14		un00 2907	. . . . . 6 |- (((A i^i { +oo}) = (/) /\ (A i^i { -oo}) = (/)) <-> ((A i^i { +oo}) u. (A i^i { -oo})) = (/))
15	12, 13, 14	3imtr3i 235	. . . . 5 |- (-. ( +oo e. A \/ -oo e. A) -> ((A i^i { +oo}) u. (A i^i { -oo})) = (/))
16		df-pr 3050	. . . . . . 7 |- { +oo, -oo} = ({ +oo} u. { -oo})
17	16	ineq2i 2793	. . . . . 6 |- (A i^i { +oo, -oo}) = (A i^i ({ +oo} u. { -oo}))
18		indi 2838	. . . . . 6 |- (A i^i ({ +oo} u. { -oo})) = ((A i^i { +oo}) u. (A i^i { -oo}))
19	17, 18	eqtri 1908	. . . . 5 |- (A i^i { +oo, -oo}) = ((A i^i { +oo}) u. (A i^i { -oo}))
20	15, 19	syl5eq 1940	. . . 4 |- (-. ( +oo e. A \/ -oo e. A) -> (A i^i { +oo, -oo}) = (/))
21	8, 20	syl5bi 225	. . 3 |- (A C_ (RR u. { +oo, -oo}) -> (-. ( +oo e. A \/ -oo e. A) -> A C_ RR))
22	21	orrd 250	. 2 |- (A C_ (RR u. { +oo, -oo}) -> (( +oo e. A \/ -oo e. A) \/ A C_ RR))
23		df-xr 6656	. . 3 |- RR* = (RR u. { +oo, -oo})
24	23	sseq2i 2642	. 2 |- (A C_ RR* <-> A C_ (RR u. { +oo, -oo}))
25		3orrot 864	. . 3 |- ((A C_ RR \/ +oo e. A \/ -oo e. A) <-> ( +oo e. A \/ -oo e. A \/ A C_ RR))
26		df-3or 859	. . 3 |- (( +oo e. A \/ -oo e. A \/ A C_ RR) <-> (( +oo e. A \/ -oo e. A) \/ A C_ RR))
27	25, 26	bitri 190	. 2 |- ((A C_ RR \/ +oo e. A \/ -oo e. A) <-> (( +oo e. A \/ -oo e. A) \/ A C_ RR))
28	22, 24, 27	3imtr4i 236	1 |- (A C_ RR* -> (A C_ RR \/ +oo e. A \/ -oo e. A))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   \/ w3o 857   = wceq 1298   e. wcel 1300   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  {cpr 3045  RRcr 6385   +oocpnf 6650   -oocmnf 6651  RR*cxr 6652

This theorem is referenced by:  xrsupss 7287  xrinfmss 7288

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-enr 6318  df-nr 6319  df-0r 6323  df-c 6392  df-r 6396  df-pnf 6654  df-mnf 6655  df-xr 6656

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