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Theorem on0eln0 6440
Description: An ordinal number contains zero iff it is nonzero. (Contributed by NM, 6-Dec-2004.)
Assertion
Ref Expression
on0eln0 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))

Proof of Theorem on0eln0
StepHypRef Expression
1 eloni 6394 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ord0eln0 6439 . 2 (Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
31, 2syl 17 1 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2108  wne 2940  c0 4333  Ord word 6383  Oncon0 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388
This theorem is referenced by:  ondif1  8539  oe0lem  8551  oevn0  8553  oa00  8597  omord  8606  om00  8613  om00el  8614  omeulem1  8620  omeulem2  8621  oewordri  8630  oeordsuc  8632  oelim2  8633  oeoa  8635  oeoe  8637  oeeui  8640  omabs  8689  omxpenlem  9113  cantnff  9714  cantnfp1  9721  cantnflem1d  9728  cantnflem1  9729  cantnflem3  9731  cantnflem4  9732  cantnf  9733  cnfcomlem  9739  cnfcom3  9744  r1tskina  10822  onsucconni  36438  onint1  36450  frlmpwfi  43110  omge1  43310  omge2  43311  omlim2  43312  omord2lim  43313  omord2i  43314  dflim5  43342  tfsconcatb0  43357  tfsconcat0b  43359  oaun3lem1  43387  naddwordnexlem4  43414  omltoe  43420
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