Theorem rpgt0d 8625

Description: A positive real is greater than zero. (Contributed by Mario Carneiro, 28-May-2016.)

Hypothesis

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

Assertion

Ref	Expression
rpgt0d	|- ( ph -> 0 < A )

Proof of Theorem rpgt0d

Step	Hyp	Ref	Expression
1		rpred.1	. 2 |- ( ph -> A e. RR+ )
2		rpgt0 8594	. 2 |- ( A e. RR+ -> 0 < A )
3	1, 2	syl 14	1 |- ( ph -> 0 < A )

Syntax hints:   -> wi 4   e. wcel 1393   class class class wbr 3764   0cc0 6889   < clt 7060   RR+crp 8583

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-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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-rp 8584

This theorem is referenced by:  rpregt0d  8629  ltmulgt11d  8658  ltmulgt12d  8659  gt0divd  8660  ge0divd  8661  lediv12ad  8682  expgt0  9288  nnesq  9368  resqrexlemp1rp  9604  resqrexlemover  9608  resqrexlemnm  9616  resqrexlemgt0  9618  resqrexlemglsq  9620  sqrtgt0d  9755  sqr2irrlem  9877