Theorem ssxr 5552

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 2330	. . . . . . . 8 |- ((A i^i { +oo, -oo}) = (/) -> (A (_ ({ +oo, -oo} u. RR) <-> A (_ RR))
2		df-xr 5501	. . . . . . . . . 10 |- RR* = (RR u. { +oo, -oo})
3		uncom 2179	. . . . . . . . . 10 |- (RR u. { +oo, -oo}) = ({ +oo, -oo} u. RR)
4	2, 3	eqtr 1498	. . . . . . . . 9 |- RR* = ({ +oo, -oo} u. RR)
5	4	sseq2i 2089	. . . . . . . 8 |- (A (_ RR* <-> A (_ ({ +oo, -oo} u. RR))
6	1, 5	syl5bb 534	. . . . . . 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 2315	. . . . . 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 1659	. . . . . 6 |- (E.x e. A x e. { +oo, -oo} <-> -. A.x e. A -. x e. { +oo, -oo})
12		visset 1816	. . . . . . . 8 |- x e. V
13	12	elpr 2428	. . . . . . 7 |- (x e. { +oo, -oo} <-> (x = +oo \/ x = -oo))
14	13	rexbii 1671	. . . . . 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 1768	. . . . 5 |- (E.x e. A (x = +oo \/ x = -oo) <-> (E.x e. A x = +oo \/ E.x e. A x = -oo))
17		df-rex 1653	. . . . . . 7 |- (E.x e. A x = +oo <-> E.x(x e. A /\ x = +oo))
18		exancom 1056	. . . . . . 7 |- (E.x(x e. A /\ x = +oo) <-> E.x(x = +oo /\ x e. A))
19		pnfxr 5505	. . . . . . . . 9 |- +oo e. RR*
20	19	elisseti 1821	. . . . . . . 8 |- +oo e. V
21		eleq1 1537	. . . . . . . 8 |- (x = +oo -> (x e. A <-> +oo e. A))
22	20, 21	ceqsexv 1838	. . . . . . 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 1653	. . . . . . 7 |- (E.x e. A x = -oo <-> E.x(x e. A /\ x = -oo))
25		exancom 1056	. . . . . . 7 |- (E.x(x e. A /\ x = -oo) <-> E.x(x = -oo /\ x e. A))
26		mnfxr 5506	. . . . . . . . 9 |- -oo e. RR*
27	26	elisseti 1821	. . . . . . . 8 |- -oo e. V
28		eleq1 1537	. . . . . . . 8 |- (x = -oo -> (x e. A <-> -oo e. A))
29	27, 28	ceqsexv 1838	. . . . . . 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 780	. 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 776   = wceq 958   e. wcel 960  E.wex 982  A.wral 1648  E.wrex 1649   u. cun 2048   i^i cin 2049   (_ wss 2050  (/)c0 2283  {cpr 2414  RRcr 5245   +oocpnf 5495   -oocmnf 5496  RR*cxr 5497

This theorem is referenced by:  xrsupss 6080  xrinfmss 6081

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-qs 4272  df-ni 5012  df-nq 5050  df-np 5098  df-nr 5179  df-c 5252  df-pnf 5499  df-mnf 5500  df-xr 5501

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