Theorem renepnfd 8125

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 8122	. 2 |- ( A e. RR -> A =/= +oo )
3	1, 2	syl 14	1 |- ( ph -> A =/= +oo )

Syntax hints:   -> wi 4   e. wcel 2176   =/= wne 2376   RRcr 7926   +oocpnf 8106

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-un 4481  ax-cnex 8018  ax-resscn 8019

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-rex 2490  df-rab 2493  df-v 2774  df-in 3172  df-ss 3179  df-pw 3618  df-uni 3851  df-pnf 8111

This theorem is referenced by:  xaddnepnf  9982  xqltnle  10412