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Theorem oninton 7202
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 7197 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
21ex 401 . . 3 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴𝐴))
3 ssel 3757 . . 3 (𝐴 ⊆ On → ( 𝐴𝐴 𝐴 ∈ On))
42, 3syld 47 . 2 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴 ∈ On))
54imp 395 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 2155  wne 2937  wss 3734  c0 4081   cint 4635  Oncon0 5910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-br 4812  df-opab 4874  df-tr 4914  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-ord 5913  df-on 5914
This theorem is referenced by:  onintrab  7203  onnmin  7205  onminex  7209  onmindif2  7214  iinon  7645  oawordeulem  7843  nnawordex  7926  tz9.12lem1  8869  rankf  8876  cardf2  9024  cff  9327  coftr  9352  sltval2  32274  nocvxminlem  32358  dnnumch3lem  38317  dnnumch3  38318
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