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Theorem onintrab 7501
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 5216 . . 3 ({𝑥 ∈ On ∣ 𝜑} ≠ ∅ ↔ {𝑥 ∈ On ∣ 𝜑} ∈ V)
2 ssrab2 4031 . . . 4 {𝑥 ∈ On ∣ 𝜑} ⊆ On
3 oninton 7500 . . . 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 238 . 2 ( {𝑥 ∈ On ∣ 𝜑} ∈ V → {𝑥 ∈ On ∣ 𝜑} ∈ On)
6 elex 3487 . 2 ( {𝑥 ∈ On ∣ 𝜑} ∈ On → {𝑥 ∈ On ∣ 𝜑} ∈ V)
75, 6impbii 212 1 ( {𝑥 ∈ On ∣ 𝜑} ∈ V ↔ {𝑥 ∈ On ∣ 𝜑} ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2114  wne 3011  {crab 3134  Vcvv 3469  wss 3908  c0 4265   cint 4851  Oncon0 6169
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173
This theorem is referenced by:  onintrab2  7502  sltval2  33237  sltres  33243
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