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Theorem ordpwsucss 4225
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 4055 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A On then both suc A = A (onunisuci 4117) and {x On ∣ xA} = A (onuniss2 4185).

Constructively (𝒫 A ∩ On) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4228). (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 4223 . . . . 5 (Ord A ↔ Ord suc A)
2 ordelon 4067 . . . . . 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 4062 . . . . 5 (Ord A → Tr A)
6 trsucss 4108 . . . . 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 3102 . . . 4 (x (𝒫 A ∩ On) ↔ (x 𝒫 A x On))
10 selpw 3340 . . . . 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 2927 1 (Ord A → suc A ⊆ (𝒫 A ∩ On))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1375  cin 2892  wss 2893  𝒫 cpw 3333  Tr wtr 3827  Ord word 4046  Oncon0 4047  suc csuc 4049
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  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 2005  ax-setind 4202
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-dif 2896  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-iord 4050  df-on 4052  df-suc 4055
This theorem is referenced by: (None)
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