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Theorem genpss 8881
 Description: The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1
genp.2
Assertion
Ref Expression
genpss
Distinct variable groups:   ,,,   ,,,   ,,,,,
Allowed substitution hints:   (,)   (,)   (,,,,)

Proof of Theorem genpss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 genp.1 . . . 4
2 genp.2 . . . 4
31, 2genpelv 8877 . . 3
4 elprnq 8868 . . . . . . . 8
54ex 424 . . . . . . 7
6 elprnq 8868 . . . . . . . 8
76ex 424 . . . . . . 7
85, 7im2anan9 809 . . . . . 6
92caovcl 6241 . . . . . 6
108, 9syl6 31 . . . . 5
11 eleq1a 2505 . . . . 5
1210, 11syl6 31 . . . 4
1312rexlimdvv 2836 . . 3
143, 13sylbid 207 . 2
1514ssrdv 3354 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cab 2422  wrex 2706   wss 3320  (class class class)co 6081   cmpt2 6083  cnq 8727  cnp 8734 This theorem is referenced by:  genpcl  8885 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-ni 8749  df-nq 8789  df-np 8858
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