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Theorem ordpwsucss 4259
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 4079 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both suc 𝐴 = 𝐴 (onunisuci 4140) and {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴 (onuniss2 4209).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4262). (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 4255 . . . . 5 (Ord 𝐴 ↔ Ord suc 𝐴)
2 ordelon 4091 . . . . . 6 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
32ex 108 . . . . 5 (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
41, 3sylbi 114 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
5 ordtr 4086 . . . . 5 (Ord 𝐴 → Tr 𝐴)
6 trsucss 4131 . . . . 5 (Tr 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
75, 6syl 14 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
84, 7jcad 291 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
9 elin 3123 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
10 selpw 3363 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1110anbi2ci 432 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
129, 11bitri 173 . . 3 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
138, 12syl6ibr 151 . 2 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ (𝒫 𝐴 ∩ On)))
1413ssrdv 2948 1 (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wcel 1393  cin 2913  wss 2914  𝒫 cpw 3356  Tr wtr 3850  Ord word 4070  Oncon0 4071  suc csuc 4073
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-in1 544  ax-in2 545  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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4230
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 2308  df-rex 2309  df-v 2556  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-uni 3577  df-tr 3851  df-iord 4074  df-on 4076  df-suc 4079
This theorem is referenced by: (None)
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