Theorem ltpnft 5542

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 1475	. . . 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 5499	. . 3 |- (A e. RR -> A e. RR*)
7		pnfxr 5493	. . . 4 |- +oo e. RR*
8		ltxrt 5495	. . . 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 696	. . 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 956   e. wcel 958   class class class wbr 2619  RRcr 5233   <R cltrr 5238   +oocpnf 5483   -oocmnf 5484  RR*cxr 5485   < clt 5486

This theorem is referenced by:  xrlttrit 5552  xrlttrt 5553  xrrebndt 5568  xrret 5569  xrinfmsslem 6077  xrub 6080  supxrre 6083  supxrunb1 6089  supxrunb2 6090  qbtwnxr 6279  elioc2t 6390  elico2t 6391  elicc2t 6392  ioomax 6393  ioopos 6394  isblo3i 8461  0bdop 9918  cdrci 10494  truni1 10499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-qs 4266  df-ni 5000  df-nq 5038  df-np 5086  df-nr 5167  df-c 5240  df-pnf 5487  df-xr 5489  df-ltxr 5490

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