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Theorem onintrab 7736
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 5299 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ {𝑥 ∈ On ∣ 𝜑} ∈ V)
2 ssrab2 4042 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 oninton 7735 . . . 4 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ {𝑥 ∈ On ∣ 𝜑} ≠ ∅) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
42, 3mpan 689 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ → {𝑥 ∈ On ∣ 𝜑} ∈ On)
51, 4sylbir 234 . 2 ( {𝑥 ∈ On ∣ 𝜑} ∈ V → {𝑥 ∈ On ∣ 𝜑} ∈ On)
6 elex 3466 . 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 2107  wne 2944  {crab 3410  Vcvv 3448  wss 3915  c0 4287   cint 4912  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  onintrab2  7737  sltval2  27020  sltres  27026
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