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Theorem onintonm 4516
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 3149 . . . . . . 7 (𝐴 ⊆ On → (𝑥𝐴𝑥 ∈ On))
2 eloni 4375 . . . . . . . 8 (𝑥 ∈ On → Ord 𝑥)
3 ordtr 4378 . . . . . . . 8 (Ord 𝑥 → Tr 𝑥)
42, 3syl 14 . . . . . . 7 (𝑥 ∈ On → Tr 𝑥)
51, 4syl6 33 . . . . . 6 (𝐴 ⊆ On → (𝑥𝐴 → Tr 𝑥))
65ralrimiv 2549 . . . . 5 (𝐴 ⊆ On → ∀𝑥𝐴 Tr 𝑥)
7 trint 4116 . . . . 5 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
86, 7syl 14 . . . 4 (𝐴 ⊆ On → Tr 𝐴)
98adantr 276 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → Tr 𝐴)
10 nfv 1528 . . . . 5 𝑥 𝐴 ⊆ On
11 nfe1 1496 . . . . 5 𝑥𝑥 𝑥𝐴
1210, 11nfan 1565 . . . 4 𝑥(𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴)
13 intssuni2m 3868 . . . . . . . 8 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 On)
14 unon 4510 . . . . . . . 8 On = On
1513, 14sseqtrdi 3203 . . . . . . 7 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ⊆ On)
1615sseld 3154 . . . . . 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 2548 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → ∀𝑥 𝐴Tr 𝑥)
20 dford3 4367 . . 3 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 𝐴Tr 𝑥))
219, 19, 20sylanbrc 417 . 2 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → Ord 𝐴)
22 inteximm 4149 . . . 4 (∃𝑥 𝑥𝐴 𝐴 ∈ V)
2322adantl 277 . . 3 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ V)
24 elong 4373 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
2523, 24syl 14 . 2 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → ( 𝐴 ∈ On ↔ Ord 𝐴))
2621, 25mpbird 167 1 ((𝐴 ⊆ On ∧ ∃𝑥 𝑥𝐴) → 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wex 1492  wcel 2148  wral 2455  Vcvv 2737  wss 3129   cuni 3809   cint 3844  Tr wtr 4101  Ord word 4362  Oncon0 4363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371
This theorem is referenced by:  onintrab2im  4517
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