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Theorem ordpwsucss 4243
 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 4074 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A ∈ On then both ∪ suc A = A (onunisuci 4135) and ∪ {x ∈ On ∣ x ⊆ A} = A (onuniss2 4203). Constructively (𝒫 A ∩ On) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4246). (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 4241 . . . . 5 (Ord A ↔ Ord suc A)
2 ordelon 4086 . . . . . 6 ((Ord suc A x 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 4081 . . . . 5 (Ord A → Tr A)
6 trsucss 4126 . . . . 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 3120 . . . 4 (x (𝒫 A ∩ On) ↔ (x 𝒫 A x On))
10 selpw 3358 . . . . 5 (x 𝒫 AxA)
1110anbi2ci 432 . . . 4 ((x 𝒫 A x 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 2945 1 (Ord A → suc A ⊆ (𝒫 A ∩ On))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390   ∩ cin 2910   ⊆ wss 2911  𝒫 cpw 3351  Tr wtr 3845  Ord word 4065  Oncon0 4066  suc csuc 4068 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074 This theorem is referenced by: (None)
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