Theorem renepnfd 8077

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 8074	. 2 |- ( A e. RR -> A =/= +oo )
3	1, 2	syl 14	1 |- ( ph -> A =/= +oo )

Syntax hints:   -> wi 4   e. wcel 2167   =/= wne 2367   RRcr 7878   +oocpnf 8058

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-un 4468  ax-cnex 7970  ax-resscn 7971

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-rex 2481  df-rab 2484  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-uni 3840  df-pnf 8063

This theorem is referenced by:  xaddnepnf  9933  xqltnle  10357