Theorem renepnfd 8003

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 8000	. 2 |- ( A e. RR -> A =/= +oo )
3	1, 2	syl 14	1 |- ( ph -> A =/= +oo )

Syntax hints:   -> wi 4   e. wcel 2148   =/= wne 2347   RRcr 7806   +oocpnf 7984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-un 4432  ax-cnex 7898  ax-resscn 7899

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-rex 2461  df-rab 2464  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-uni 3810  df-pnf 7989

This theorem is referenced by:  xaddnepnf  9853