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Theorem ordpwsucss 4541
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 4346 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both suc 𝐴 = 𝐴 (onunisuci 4407) and {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴 (onuniss2 4486).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4544). (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 4537 . . . . 5 (Ord 𝐴 ↔ Ord suc 𝐴)
2 ordelon 4358 . . . . . 6 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
32ex 114 . . . . 5 (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
41, 3sylbi 120 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
5 ordtr 4353 . . . . 5 (Ord 𝐴 → Tr 𝐴)
6 trsucss 4398 . . . . 5 (Tr 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
75, 6syl 14 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
84, 7jcad 305 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
9 elin 3303 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
10 velpw 3563 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1110anbi2ci 455 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
129, 11bitri 183 . . 3 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
138, 12syl6ibr 161 . 2 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ (𝒫 𝐴 ∩ On)))
1413ssrdv 3146 1 (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2135  cin 3113  wss 3114  𝒫 cpw 3556  Tr wtr 4077  Ord word 4337  Oncon0 4338  suc csuc 4340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-setind 4511
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2726  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-uni 3787  df-tr 4078  df-iord 4341  df-on 4343  df-suc 4346
This theorem is referenced by: (None)
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