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Theorem oninton 7782
Description: The intersection of a nonempty collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by NM, 29-Jan-1997.)
Assertion
Ref Expression
oninton ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)

Proof of Theorem oninton
StepHypRef Expression
1 onint 7777 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
21ex 413 . . 3 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴𝐴))
3 ssel 3975 . . 3 (𝐴 ⊆ On → ( 𝐴𝐴 𝐴 ∈ On))
42, 3syld 47 . 2 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴 ∈ On))
54imp 407 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wne 2940  wss 3948  c0 4322   cint 4950  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  onintrab  7783  onnmin  7785  onminex  7789  onmindif2  7794  iinon  8339  oawordeulem  8553  nnawordex  8636  tz9.12lem1  9781  rankf  9788  cardf2  9937  cff  10242  coftr  10267  sltval2  27156  nocvxminlem  27276  dnnumch3lem  41778  dnnumch3  41779  onintunirab  41966  oninfint  41975  oninfcl2  41977  naddwordnexlem4  42142
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