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Theorem onintrab2im 4244
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 3025 . 2 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfrab1 2489 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
32nfcri 2172 . . . 4 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
43nfex 1528 . . 3 𝑥𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
5 rabid 2485 . . . . 5 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
6 elex2 2570 . . . . 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 2426 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
10 onintonm 4243 . 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 2307  {crab 2310  wss 2917   cint 3615  Oncon0 4100
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 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
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 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108
This theorem is referenced by:  cardcl  6359
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