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Theorem oninton 7507
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 7502 . . . 4 ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → 𝐴𝐴)
21ex 415 . . 3 (𝐴 ⊆ On → (𝐴 ≠ ∅ → 𝐴𝐴))
3 ssel 3959 . . 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 2108  wne 3014  wss 3934  c0 4289   cint 4867  Oncon0 6184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188
This theorem is referenced by:  onintrab  7508  onnmin  7510  onminex  7514  onmindif2  7519  iinon  7969  oawordeulem  8172  nnawordex  8255  tz9.12lem1  9208  rankf  9215  cardf2  9364  cff  9662  coftr  9687  sltval2  33156  nocvxminlem  33240  dnnumch3lem  39637  dnnumch3  39638
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