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Theorem inisegn0 27156
 Description: Non-emptyness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Assertion
Ref Expression
inisegn0

Proof of Theorem inisegn0
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2970 . 2
2 snprc 3895 . . . . . 6
32biimpi 188 . . . . 5
43imaeq2d 5232 . . . 4
5 ima0 5250 . . . 4
64, 5syl6eq 2490 . . 3
76necon1ai 2652 . 2
8 eleq1 2502 . . 3
9 sneq 3849 . . . . 5
109imaeq2d 5232 . . . 4
1110neeq1d 2620 . . 3
12 abn0 3631 . . . 4
13 vex 2965 . . . . . 6
14 iniseg 5264 . . . . . 6
1513, 14ax-mp 5 . . . . 5
1615neeq1i 2617 . . . 4
1713elrn 5139 . . . 4
1812, 16, 173bitr4ri 271 . . 3
198, 11, 18vtoclbg 3018 . 2
201, 7, 19pm5.21nii 344 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 178  wex 1551   wceq 1653   wcel 1727  cab 2428   wne 2605  cvv 2962  c0 3613  csn 3838   class class class wbr 4237  ccnv 4906   crn 4908  cima 4910 This theorem is referenced by:  dnnumch3lem  27159  dnnumch3  27160 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-xp 4913  df-cnv 4915  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920
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