Theorem ltpnft 5515

Description: Any (finite) real is less than plus infinity.

Assertion

Ref	Expression
ltpnft	|- (A e. RR -> A < +oo)

Proof of Theorem ltpnft

Step	Hyp	Ref	Expression
1		eqid 1468	. . . 4 |- +oo = +oo
2	1	jctr 291	. . 3 |- (A e. RR -> (A e. RR /\ +oo = +oo))
3		orc 269	. . 3 |- ((A e. RR /\ +oo = +oo) -> ((A e. RR /\ +oo = +oo) \/ (A = -oo /\ +oo e. RR)))
4		olc 268	. . 3 |- (((A e. RR /\ +oo = +oo) \/ (A = -oo /\ +oo e. RR)) -> ((((A e. RR /\ +oo e. RR) /\ A <R +oo) \/ (A = -oo /\ +oo = +oo)) \/ ((A e. RR /\ +oo = +oo) \/ (A = -oo /\ +oo e. RR))))
5	2, 3, 4	3syl 20	. 2 |- (A e. RR -> ((((A e. RR /\ +oo e. RR) /\ A <R +oo) \/ (A = -oo /\ +oo = +oo)) \/ ((A e. RR /\ +oo = +oo) \/ (A = -oo /\ +oo e. RR))))
6		rexrt 5471	. . 3 |- (A e. RR -> A e. RR*)
7		pnfxr 5465	. . . 4 |- +oo e. RR*
8		ltxrt 5467	. . . 4 |- ((A e. RR* /\ +oo e. RR*) -> (A < +oo <-> ((((A e. RR /\ +oo e. RR) /\ A <R +oo) \/ (A = -oo /\ +oo = +oo)) \/ ((A e. RR /\ +oo = +oo) \/ (A = -oo /\ +oo e. RR)))))
9	7, 8	mpan2 694	. . 3 |- (A e. RR* -> (A < +oo <-> ((((A e. RR /\ +oo e. RR) /\ A <R +oo) \/ (A = -oo /\ +oo = +oo)) \/ ((A e. RR /\ +oo = +oo) \/ (A = -oo /\ +oo e. RR)))))
10	6, 9	syl 10	. 2 |- (A e. RR -> (A < +oo <-> ((((A e. RR /\ +oo e. RR) /\ A <R +oo) \/ (A = -oo /\ +oo = +oo)) \/ ((A e. RR /\ +oo = +oo) \/ (A = -oo /\ +oo e. RR)))))
11	5, 10	mpbird 196	1 |- (A e. RR -> A < +oo)

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   class class class wbr 2609  RRcr 5205   <R cltrr 5210   +oocpnf 5455   -oocmnf 5456  RR*cxr 5457   < clt 5458

This theorem is referenced by:  xrlttrit 5525  xrlttrt 5526  xrrebndt 5541  xrret 5542  xrinfmsslem 6024  xrub 6027  supxrre 6030  supxrunb1 6036  supxrunb2 6037  qbtwnxr 6217  elioc2t 6322  elico2t 6323  elicc2t 6324  ioomax 6325  ioopos 6326  isblo3i 8392  0bdop 9833  cdrci 10381  truni1 10386

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-xr 5461  df-ltxr 5462

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