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Theorem onintonm 4243
 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 2939 . . . . . . 7 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
2 eloni 4112 . . . . . . . 8 (𝑥 ∈ On → Ord 𝑥)
3 ordtr 4115 . . . . . . . 8 (Ord 𝑥 → Tr 𝑥)
42, 3syl 14 . . . . . . 7 (𝑥 ∈ On → Tr 𝑥)
51, 4syl6 29 . . . . . 6 (𝐴 ⊆ On → (𝑥𝐴 → Tr 𝑥))
65ralrimiv 2391 . . . . 5 (𝐴 ⊆ On → ∀𝑥𝐴 Tr 𝑥)
7 trint 3869 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
86, 7syl 14 . . . 4 (𝐴 ⊆ On → Tr 𝐴)
98adantr 261 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → Tr 𝐴)
10 nfv 1421 . . . . 5 𝑥 𝐴 ⊆ On
11 nfe1 1385 . . . . 5 𝑥𝑥 𝑥𝐴
1210, 11nfan 1457 . . . 4 𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴)
13 intssuni2m 3639 . . . . . . . 8 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 On)
14 unon 4237 . . . . . . . 8 On = On
1513, 14syl6sseq 2991 . . . . . . 7 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ⊆ On)
1615sseld 2944 . . . . . 6 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → (𝑥 𝐴𝑥 ∈ On))
1716, 2syl6 29 . . . . 5 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → (𝑥 𝐴 → Ord 𝑥))
1817, 3syl6 29 . . . 4 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → (𝑥 𝐴 → Tr 𝑥))
1912, 18ralrimi 2390 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → ∀𝑥 𝐴Tr 𝑥)
20 dford3 4104 . . 3 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 𝐴Tr 𝑥))
219, 19, 20sylanbrc 394 . 2 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → Ord 𝐴)
22 inteximm 3903 . . . 4 (∃𝑥 𝑥𝐴 𝐴 ∈ V)
2322adantl 262 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ V)
24 elong 4110 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
2523, 24syl 14 . 2 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → ( 𝐴 ∈ On ↔ Ord 𝐴))
2621, 25mpbird 156 1 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  Vcvv 2557   ⊆ wss 2917  ∪ cuni 3580  ∩ cint 3615  Tr wtr 3854  Ord word 4099  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-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:  onintrab2im  4244
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