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Theorem onintrab2im 4216
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 3022 . 2 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfrab1 2486 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
32nfcri 2172 . . . 4 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
43nfex 1528 . . 3 𝑥𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
5 rabid 2482 . . . . 5 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
6 elex2 2567 . . . . 5 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
75, 6sylbir 125 . . . 4 ((𝑥 ∈ On ∧ 𝜑) → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
87ex 108 . . 3 (𝑥 ∈ On → (𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}))
94, 8rexlimi 2423 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
10 onintonm 4215 . 2 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
111, 9, 10sylancr 393 1 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wex 1381  wcel 1393  wrex 2304  {crab 2307  wss 2914   cint 3612  Oncon0 4072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4142
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-rab 2312  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-uni 3578  df-int 3613  df-tr 3852  df-iord 4075  df-on 4077  df-suc 4080
This theorem is referenced by:  cardcl  6318
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