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Theorem onintrab2im 4502
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 3232 . 2 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfrab1 2649 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
32nfcri 2306 . . . 4 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
43nfex 1630 . . 3 𝑥𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
5 rabid 2645 . . . . 5 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
6 elex2 2746 . . . . 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 2580 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
10 onintonm 4501 . 2 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
111, 9, 10sylancr 412 1 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1485  wcel 2141  wrex 2449  {crab 2452  wss 3121   cint 3831  Oncon0 4348
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356
This theorem is referenced by:  cardcl  7158
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