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Theorem oninton 6954
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 6949 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
21ex 450 . . 3 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴𝐴))
3 ssel 3581 . . 3 (𝐴 ⊆ On → ( 𝐴𝐴 𝐴 ∈ On))
42, 3syld 47 . 2 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴 ∈ On))
54imp 445 1 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wne 2790  wss 3559  c0 3896   cint 4445  Oncon0 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690  df-on 5691
This theorem is referenced by:  onintrab  6955  onnmin  6957  onminex  6961  onmindif2  6966  iinon  7389  oawordeulem  7586  nnawordex  7669  tz9.12lem1  8602  rankf  8609  cardf2  8721  cff  9022  coftr  9047  sltval2  31545  nodenselem4  31582  nocvxminlem  31588  dnnumch3lem  37131  dnnumch3  37132
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