Theorem renepnfd 7169

Description: No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Hypothesis

Ref	Expression
rexrd.1	|- ( ph -> A e. RR )

Assertion

Ref	Expression
renepnfd	|- ( ph -> A =/= +oo )

Proof of Theorem renepnfd

Step	Hyp	Ref	Expression
1		rexrd.1	. 2 |- ( ph -> A e. RR )
2		renepnf 7166	. 2 |- ( A e. RR -> A =/= +oo )
3	1, 2	syl 14	1 |- ( ph -> A =/= +oo )

Syntax hints:   -> wi 4   e. wcel 1433   =/= wne 2245   RRcr 6980   +oocpnf 7150

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-un 4188  ax-cnex 7067  ax-resscn 7068

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-rex 2354  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-uni 3602  df-pnf 7155

This theorem is referenced by: (None)