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Theorem onintrab 7780
Description: The intersection of a class of ordinal numbers exists iff it is an ordinal number. (Contributed by NM, 6-Nov-2003.)
Assertion
Ref Expression
onintrab ( {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ {𝑥 ∈ On ∣ 𝜑} ∈ On)

Proof of Theorem onintrab
StepHypRef Expression
1 intex 5336 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ {𝑥 ∈ On ∣ 𝜑} ∈ V)
2 ssrab2 4076 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 oninton 7779 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
42, 3mpan 688 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → {𝑥 ∈ On ∣ 𝜑} ∈ On)
51, 4sylbir 234 . 2 ( {𝑥 ∈ On ∣ 𝜑} ∈ V → {𝑥 ∈ On ∣ 𝜑} ∈ On)
6 elex 3492 . 2 ( {𝑥 ∈ On ∣ 𝜑} ∈ On → {𝑥 ∈ On ∣ 𝜑} ∈ V)
75, 6impbii 208 1 ( {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ {𝑥 ∈ On ∣ 𝜑} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2106  wne 2940  {crab 3432  Vcvv 3474  wss 3947  c0 4321   cint 4949  Oncon0 6361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365
This theorem is referenced by:  onintrab2  7781  sltval2  27148  sltres  27154
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