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Theorem onintonm 4300
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 3006 . . . . . . 7 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
2 eloni 4169 . . . . . . . 8 (𝑥 ∈ On → Ord 𝑥)
3 ordtr 4172 . . . . . . . 8 (Ord 𝑥 → Tr 𝑥)
42, 3syl 14 . . . . . . 7 (𝑥 ∈ On → Tr 𝑥)
51, 4syl6 33 . . . . . 6 (𝐴 ⊆ On → (𝑥𝐴 → Tr 𝑥))
65ralrimiv 2441 . . . . 5 (𝐴 ⊆ On → ∀𝑥𝐴 Tr 𝑥)
7 trint 3919 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
86, 7syl 14 . . . 4 (𝐴 ⊆ On → Tr 𝐴)
98adantr 270 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → Tr 𝐴)
10 nfv 1464 . . . . 5 𝑥 𝐴 ⊆ On
11 nfe1 1428 . . . . 5 𝑥𝑥 𝑥𝐴
1210, 11nfan 1500 . . . 4 𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴)
13 intssuni2m 3689 . . . . . . . 8 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 On)
14 unon 4294 . . . . . . . 8 On = On
1513, 14syl6sseq 3058 . . . . . . 7 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ⊆ On)
1615sseld 3011 . . . . . 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 2440 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → ∀𝑥 𝐴Tr 𝑥)
20 dford3 4161 . . 3 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 𝐴Tr 𝑥))
219, 19, 20sylanbrc 408 . 2 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → Ord 𝐴)
22 inteximm 3953 . . . 4 (∃𝑥 𝑥𝐴 𝐴 ∈ V)
2322adantl 271 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ V)
24 elong 4167 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
2523, 24syl 14 . 2 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → ( 𝐴 ∈ On ↔ Ord 𝐴))
2621, 25mpbird 165 1 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1424  wcel 1436  wral 2355  Vcvv 2614  wss 2986   cuni 3630   cint 3665  Tr wtr 3904  Ord word 4156  Oncon0 4157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-uni 3631  df-int 3666  df-tr 3905  df-iord 4160  df-on 4162  df-suc 4165
This theorem is referenced by:  onintrab2im  4301
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