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Theorem ordpwsucss 4196
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 4031 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A On then both suc A = A (onunisuci 4092) and {x On ∣ xA} = A (onuniss2 4160).

Constructively (𝒫 A ∩ On) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4199). (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 4195 . . . . 5 (Ord A ↔ Ord suc A)
2 ordelon 4043 . . . . . 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 4038 . . . . 5 (Ord A → Tr A)
6 trsucss 4083 . . . . 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 3104 . . . 4 (x (𝒫 A ∩ On) ↔ (x 𝒫 A x On))
10 selpw 3318 . . . . 5 (x 𝒫 AxA)
1110anbi2ci 435 . . . 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 2929 1 (Ord A → suc A ⊆ (𝒫 A ∩ On))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1375  cin 2894  wss 2895  𝒫 cpw 3311  Tr wtr 3806  Ord word 4023  Oncon0 4024  suc csuc 4026
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 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-setind 4174
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-dif 2898  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-uni 3533  df-tr 3807  df-iord 4027  df-on 4028  df-suc 4031
This theorem is referenced by: (None)
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