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Theorem oninton 7509
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 7504 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
21ex 415 . . 3 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴𝐴))
3 ssel 3960 . . 3 (𝐴 ⊆ On → ( 𝐴𝐴 𝐴 ∈ On))
42, 3syld 47 . 2 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴 ∈ On))
54imp 409 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2110  wne 3016  wss 3935  c0 4290   cint 4868  Oncon0 6185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188  df-on 6189
This theorem is referenced by:  onintrab  7510  onnmin  7512  onminex  7516  onmindif2  7521  iinon  7971  oawordeulem  8174  nnawordex  8257  tz9.12lem1  9210  rankf  9217  cardf2  9366  cff  9664  coftr  9689  sltval2  33158  nocvxminlem  33242  dnnumch3lem  39639  dnnumch3  39640
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