Theorem ssxr 5513

Description: The three (non-exclusive) possibilities implied by a subset of extended reals.

Assertion

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

Proof of Theorem ssxr

Step	Hyp	Ref	Expression
1		disjssun 2316	. . . . . . . 8 |- ((A i^i { +oo, -oo}) = (/) -> (A (_ ({ +oo, -oo} u. RR) <-> A (_ RR))
2		df-xr 5461	. . . . . . . . . 10 |- RR* = (RR u. { +oo, -oo})
3		uncom 2166	. . . . . . . . . 10 |- (RR u. { +oo, -oo}) = ({ +oo, -oo} u. RR)
4	2, 3	eqtr 1487	. . . . . . . . 9 |- RR* = ({ +oo, -oo} u. RR)
5	4	sseq2i 2076	. . . . . . . 8 |- (A (_ RR* <-> A (_ ({ +oo, -oo} u. RR))
6	1, 5	syl5bb 530	. . . . . . 7 |- ((A i^i { +oo, -oo}) = (/) -> (A (_ RR* <-> A (_ RR))
7	6	biimpcd 155	. . . . . 6 |- (A (_ RR* -> ((A i^i { +oo, -oo}) = (/) -> A (_ RR))
8		disj 2301	. . . . . 6 |- ((A i^i { +oo, -oo}) = (/) <-> A.x e. A -. x e. { +oo, -oo})
9	7, 8	syl5ibr 207	. . . . 5 |- (A (_ RR* -> (A.x e. A -. x e. { +oo, -oo} -> A (_ RR))
10	9	con3d 95	. . . 4 |- (A (_ RR* -> (-. A (_ RR -> -. A.x e. A -. x e. { +oo, -oo}))
11		dfrex2 1648	. . . . . 6 |- (E.x e. A x e. { +oo, -oo} <-> -. A.x e. A -. x e. { +oo, -oo})
12		visset 1804	. . . . . . . 8 |- x e. V
13	12	elpr 2414	. . . . . . 7 |- (x e. { +oo, -oo} <-> (x = +oo \/ x = -oo))
14	13	rexbii 1660	. . . . . 6 |- (E.x e. A x e. { +oo, -oo} <-> E.x e. A (x = +oo \/ x = -oo))
15	11, 14	bitr3 175	. . . . 5 |- (-. A.x e. A -. x e. { +oo, -oo} <-> E.x e. A (x = +oo \/ x = -oo))
16		r19.43 1757	. . . . 5 |- (E.x e. A (x = +oo \/ x = -oo) <-> (E.x e. A x = +oo \/ E.x e. A x = -oo))
17		df-rex 1642	. . . . . . 7 |- (E.x e. A x = +oo <-> E.x(x e. A /\ x = +oo))
18		exancom 1050	. . . . . . 7 |- (E.x(x e. A /\ x = +oo) <-> E.x(x = +oo /\ x e. A))
19		pnfxr 5465	. . . . . . . . 9 |- +oo e. RR*
20	19	elisseti 1809	. . . . . . . 8 |- +oo e. V
21		eleq1 1526	. . . . . . . 8 |- (x = +oo -> (x e. A <-> +oo e. A))
22	20, 21	ceqsexv 1826	. . . . . . 7 |- (E.x(x = +oo /\ x e. A) <-> +oo e. A)
23	17, 18, 22	3bitr 177	. . . . . 6 |- (E.x e. A x = +oo <-> +oo e. A)
24		df-rex 1642	. . . . . . 7 |- (E.x e. A x = -oo <-> E.x(x e. A /\ x = -oo))
25		exancom 1050	. . . . . . 7 |- (E.x(x e. A /\ x = -oo) <-> E.x(x = -oo /\ x e. A))
26		mnfxr 5466	. . . . . . . . 9 |- -oo e. RR*
27	26	elisseti 1809	. . . . . . . 8 |- -oo e. V
28		eleq1 1526	. . . . . . . 8 |- (x = -oo -> (x e. A <-> -oo e. A))
29	27, 28	ceqsexv 1826	. . . . . . 7 |- (E.x(x = -oo /\ x e. A) <-> -oo e. A)
30	24, 25, 29	3bitr 177	. . . . . 6 |- (E.x e. A x = -oo <-> -oo e. A)
31	23, 30	orbi12i 257	. . . . 5 |- ((E.x e. A x = +oo \/ E.x e. A x = -oo) <-> ( +oo e. A \/ -oo e. A))
32	15, 16, 31	3bitr 177	. . . 4 |- (-. A.x e. A -. x e. { +oo, -oo} <-> ( +oo e. A \/ -oo e. A))
33	10, 32	syl6ib 212	. . 3 |- (A (_ RR* -> (-. A (_ RR -> ( +oo e. A \/ -oo e. A)))
34	33	orrd 233	. 2 |- (A (_ RR* -> (A (_ RR \/ ( +oo e. A \/ -oo e. A)))
35		3orass 776	. 2 |- ((A (_ RR \/ +oo e. A \/ -oo e. A) <-> (A (_ RR \/ ( +oo e. A \/ -oo e. A)))
36	34, 35	sylibr 200	1 |- (A (_ RR* -> (A (_ RR \/ +oo e. A \/ -oo e. A))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955  E.wex 977  A.wral 1637  E.wrex 1638   u. cun 2035   i^i cin 2036   (_ wss 2037  (/)c0 2270  {cpr 2400  RRcr 5205   +oocpnf 5455   -oocmnf 5456  RR*cxr 5457

This theorem is referenced by:  xrsupss 6025  xrinfmss 6026

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-qs 4250  df-ni 4972  df-nq 5010  df-np 5058  df-nr 5139  df-c 5212  df-pnf 5459  df-mnf 5460  df-xr 5461

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