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Theorem on0eln0 5818
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 5771 . 2 (𝐴 ∈ On → Ord 𝐴)
2 ord0eln0 5817 . 2 (Ord 𝐴 → (∅ ∈ 𝐴𝐴 ≠ ∅))
31, 2syl 17 1 (𝐴 ∈ On → (∅ ∈ 𝐴𝐴 ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2030  wne 2823  c0 3948  Ord word 5760  Oncon0 5761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765
This theorem is referenced by:  ondif1  7626  oe0lem  7638  oevn0  7640  oa00  7684  omord  7693  om00  7700  om00el  7701  omeulem1  7707  omeulem2  7708  oewordri  7717  oeordsuc  7719  oelim2  7720  oeoa  7722  oeoe  7724  oeeui  7727  omabs  7772  omxpenlem  8102  cantnff  8609  cantnfp1lem2  8614  cantnfp1lem3  8615  cantnfp1  8616  cantnflem1d  8623  cantnflem1  8624  cantnflem3  8626  cantnflem4  8627  cantnf  8628  cnfcomlem  8634  cnfcom3  8639  r1tskina  9642  onsucconni  32561  onint1  32573  frlmpwfi  37985
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