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Theorem ordpwsucss 4345
Description: The collection of ordinals in the power class of an ordinal is a superset of its successor.

We can think of (𝒫 𝐴 ∩ On) as another possible definition of successor, which would be equivalent to df-suc 4161 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both suc 𝐴 = 𝐴 (onunisuci 4222) and {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴 (onuniss2 4291).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4348). (Contributed by Jim Kingdon, 21-Jul-2019.)

Assertion
Ref Expression
ordpwsucss (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))

Proof of Theorem ordpwsucss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordsuc 4341 . . . . 5 (Ord 𝐴 ↔ Ord suc 𝐴)
2 ordelon 4173 . . . . . 6 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
32ex 113 . . . . 5 (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
41, 3sylbi 119 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
5 ordtr 4168 . . . . 5 (Ord 𝐴 → Tr 𝐴)
6 trsucss 4213 . . . . 5 (Tr 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
75, 6syl 14 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
84, 7jcad 301 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
9 elin 3167 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
10 selpw 3413 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1110anbi2ci 447 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
129, 11bitri 182 . . 3 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
138, 12syl6ibr 160 . 2 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ (𝒫 𝐴 ∩ On)))
1413ssrdv 3016 1 (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1434  cin 2983  wss 2984  𝒫 cpw 3406  Tr wtr 3901  Ord word 4152  Oncon0 4153  suc csuc 4155
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-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-setind 4315
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-tr 3902  df-iord 4156  df-on 4158  df-suc 4161
This theorem is referenced by: (None)
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