Theorem renepnfd 7076

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 7073	. 2 |- ( A e. RR -> A =/= +oo )
3	1, 2	syl 14	1 |- ( ph -> A =/= +oo )

Syntax hints:   -> wi 4   e. wcel 1393   =/= wne 2204   RRcr 6888   +oocpnf 7057

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-un 4170  ax-cnex 6975  ax-resscn 6976

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581  df-pnf 7062

This theorem is referenced by: (None)