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Theorem onintonm 4371
Description: The intersection of an inhabited collection of ordinal numbers is an ordinal number. Compare Exercise 6 of [TakeutiZaring] p. 44. (Contributed by Mario Carneiro and Jim Kingdon, 30-Aug-2021.)
Assertion
Ref Expression
onintonm ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ On)
Distinct variable group:   𝑥,𝐴

Proof of Theorem onintonm
StepHypRef Expression
1 ssel 3041 . . . . . . 7 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
2 eloni 4235 . . . . . . . 8 (𝑥 ∈ On → Ord 𝑥)
3 ordtr 4238 . . . . . . . 8 (Ord 𝑥 → Tr 𝑥)
42, 3syl 14 . . . . . . 7 (𝑥 ∈ On → Tr 𝑥)
51, 4syl6 33 . . . . . 6 (𝐴 ⊆ On → (𝑥𝐴 → Tr 𝑥))
65ralrimiv 2463 . . . . 5 (𝐴 ⊆ On → ∀𝑥𝐴 Tr 𝑥)
7 trint 3981 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
86, 7syl 14 . . . 4 (𝐴 ⊆ On → Tr 𝐴)
98adantr 272 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → Tr 𝐴)
10 nfv 1476 . . . . 5 𝑥 𝐴 ⊆ On
11 nfe1 1440 . . . . 5 𝑥𝑥 𝑥𝐴
1210, 11nfan 1512 . . . 4 𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴)
13 intssuni2m 3742 . . . . . . . 8 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 On)
14 unon 4365 . . . . . . . 8 On = On
1513, 14syl6sseq 3095 . . . . . . 7 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ⊆ On)
1615sseld 3046 . . . . . 6 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → (𝑥 𝐴𝑥 ∈ On))
1716, 2syl6 33 . . . . 5 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → (𝑥 𝐴 → Ord 𝑥))
1817, 3syl6 33 . . . 4 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → (𝑥 𝐴 → Tr 𝑥))
1912, 18ralrimi 2462 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → ∀𝑥 𝐴Tr 𝑥)
20 dford3 4227 . . 3 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 𝐴Tr 𝑥))
219, 19, 20sylanbrc 411 . 2 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → Ord 𝐴)
22 inteximm 4014 . . . 4 (∃𝑥 𝑥𝐴 𝐴 ∈ V)
2322adantl 273 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ V)
24 elong 4233 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
2523, 24syl 14 . 2 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → ( 𝐴 ∈ On ↔ Ord 𝐴))
2621, 25mpbird 166 1 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wex 1436  wcel 1448  wral 2375  Vcvv 2641  wss 3021   cuni 3683   cint 3718  Tr wtr 3966  Ord word 4222  Oncon0 4223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-tr 3967  df-iord 4226  df-on 4228  df-suc 4231
This theorem is referenced by:  onintrab2im  4372
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