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Theorem onintrab2 7798
Description: An existence condition equivalent to an intersection's being an ordinal number. (Contributed by NM, 6-Nov-2003.)
Assertion
Ref Expression
onintrab2 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)

Proof of Theorem onintrab2
StepHypRef Expression
1 intexrab 5337 . 2 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ V)
2 onintrab 7797 . 2 ( {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ {𝑥 ∈ On ∣ 𝜑} ∈ On)
31, 2bitri 274 1 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  wrex 3060  {crab 3419  Vcvv 3463   cint 4944  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368
This theorem is referenced by:  oeeulem  8620  cofon1  8691  cofon2  8692  cofonr  8693  naddcllem  8695  naddunif  8712  cardmin2  10022  cardaleph  10112  cardmin  10587  nosepon  27616  minregex  43029
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