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Theorem onintrab2im 4477
Description: An existence condition which implies an intersection is an ordinal number. (Contributed by Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
onintrab2im (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)

Proof of Theorem onintrab2im
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3213 . 2 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfrab1 2636 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
32nfcri 2293 . . . 4 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
43nfex 1617 . . 3 𝑥𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
5 rabid 2632 . . . . 5 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
6 elex2 2728 . . . . 5 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
75, 6sylbir 134 . . . 4 ((𝑥 ∈ On ∧ 𝜑) → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
87ex 114 . . 3 (𝑥 ∈ On → (𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}))
94, 8rexlimi 2567 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
10 onintonm 4476 . 2 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
111, 9, 10sylancr 411 1 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1472  wcel 2128  wrex 2436  {crab 2439  wss 3102   cint 3807  Oncon0 4323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331
This theorem is referenced by:  cardcl  7116
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