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Theorem ordpwsucss 4227
 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 4057 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A ∈ On then both ∪ suc A = A (onunisuci 4119) and ∪ {x ∈ On ∣ x ⊆ A} = A (onuniss2 4187). Constructively (𝒫 A ∩ On) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4230). (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 4225 . . . . 5 (Ord A ↔ Ord suc A)
2 ordelon 4069 . . . . . 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 4064 . . . . 5 (Ord A → Tr A)
6 trsucss 4110 . . . . 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 3103 . . . 4 (x (𝒫 A ∩ On) ↔ (x 𝒫 A x On))
10 selpw 3341 . . . . 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 2928 1 (Ord A → suc A ⊆ (𝒫 A ∩ On))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1374   ∩ cin 2893   ⊆ wss 2894  𝒫 cpw 3334  Tr wtr 3828  Ord word 4048  Oncon0 4049  suc csuc 4051 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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057 This theorem is referenced by: (None)
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