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Theorem shne0i 22955
 Description: A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
shne0.1
Assertion
Ref Expression
shne0i
Distinct variable group:   ,

Proof of Theorem shne0i
StepHypRef Expression
1 df-ne 2603 . 2
2 df-rex 2713 . . 3
3 nss 3408 . . 3
4 shne0.1 . . . . 5
5 shle0 22949 . . . . 5
64, 5ax-mp 5 . . . 4
76notbii 289 . . 3
82, 3, 73bitr2ri 267 . 2
9 elch0 22761 . . . 4
109necon3bbii 2634 . . 3
1110rexbii 2732 . 2
121, 8, 113bitri 264 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726   wne 2601  wrex 2708   wss 3322  c0v 22432  csh 22436  c0h 22443 This theorem is referenced by:  chne0i  22960  shatomici  23866 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 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-hilex 22507  ax-hv0cl 22511 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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-sh 22714  df-ch0 22760
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