Theorem ltpnf 9737

Description: Any (finite) real is less than plus infinity. (Contributed by NM, 14-Oct-2005.)

Assertion

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

Proof of Theorem ltpnf

Step	Hyp	Ref	Expression
1		eqid 2170	. . . 4 |- +oo = +oo
2		orc 707	. . . 4 |- ( ( A e. RR /\ +oo = +oo ) -> ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) )
3	1, 2	mpan2 423	. . 3 |- ( A e. RR -> ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) )
4	3	olcd 729	. 2 |- ( A e. RR -> ( ( ( ( A e. RR /\ +oo e. RR ) /\ A <RR +oo ) \/ ( A = -oo /\ +oo = +oo ) ) \/ ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) ) )
5		rexr 7965	. . 3 |- ( A e. RR -> A e. RR* )
6		pnfxr 7972	. . 3 |- +oo e. RR*
7		ltxr 9732	. . 3 |- ( ( A e. RR* /\ +oo e. RR* ) -> ( A < +oo <-> ( ( ( ( A e. RR /\ +oo e. RR ) /\ A <RR +oo ) \/ ( A = -oo /\ +oo = +oo ) ) \/ ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) ) ) )
8	5, 6, 7	sylancl 411	. 2 |- ( A e. RR -> ( A < +oo <-> ( ( ( ( A e. RR /\ +oo e. RR ) /\ A <RR +oo ) \/ ( A = -oo /\ +oo = +oo ) ) \/ ( ( A e. RR /\ +oo = +oo ) \/ ( A = -oo /\ +oo e. RR ) ) ) ) )
9	4, 8	mpbird 166	1 |- ( A e. RR -> A < +oo )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   class class class wbr 3989   RRcr 7773   <RR cltrr 7778   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953   < clt 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-cnex 7865

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-xr 7958  df-ltxr 7959

This theorem is referenced by:  ltpnfd  9738  0ltpnf  9739  xrlttr  9752  xrltso  9753  xrlttri3  9754  nltpnft  9771  npnflt  9772  xrrebnd  9776  xrre  9777  xltnegi  9792  xltadd1  9833  xposdif  9839  elioc2  9893  elicc2  9895  ioomax  9905  ioopos  9907  elioopnf  9924  elicopnf  9926  qbtwnxr  10214  dfrp2  10220  filtinf  10726  xrmaxltsup  11221  fprodge0  11600  fprodge1  11602  xblss2ps  13198