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Theorem onintrab2im 4272
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 3053 . 2 {𝑥 ∈ On ∣ 𝜑} ⊆ On
2 nfrab1 2506 . . . . 5 𝑥{𝑥 ∈ On ∣ 𝜑}
32nfcri 2188 . . . 4 𝑥 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
43nfex 1544 . . 3 𝑥𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}
5 rabid 2502 . . . . 5 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} ↔ (𝑥 ∈ On ∧ 𝜑))
6 elex2 2587 . . . . 5 (𝑥 ∈ {𝑥 ∈ On ∣ 𝜑} → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
75, 6sylbir 129 . . . 4 ((𝑥 ∈ On ∧ 𝜑) → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
87ex 112 . . 3 (𝑥 ∈ On → (𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}))
94, 8rexlimi 2443 . 2 (∃𝑥 ∈ On 𝜑 → ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑})
10 onintonm 4271 . 2 (({𝑥 ∈ On ∣ 𝜑} ⊆ On ∧ ∃𝑦 𝑦 ∈ {𝑥 ∈ On ∣ 𝜑}) → {𝑥 ∈ On ∣ 𝜑} ∈ On)
111, 9, 10sylancr 399 1 (∃𝑥 ∈ On 𝜑 {𝑥 ∈ On ∣ 𝜑} ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wex 1397  wcel 1409  wrex 2324  {crab 2327  wss 2945   cint 3643  Oncon0 4128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by:  cardcl  6419
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