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Theorem ordpwsucss 4411
 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 4222 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4283) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4357). Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4414). (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 4407 . . . . 5 (Ord 𝐴 ↔ Ord suc 𝐴)
2 ordelon 4234 . . . . . 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 4229 . . . . 5 (Ord 𝐴 → Tr 𝐴)
6 trsucss 4274 . . . . 5 (Tr 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
75, 6syl 14 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
84, 7jcad 302 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
9 elin 3198 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
10 selpw 3456 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1110anbi2ci 448 . . . 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 3045 1 (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1445   ∩ cin 3012   ⊆ wss 3013  𝒫 cpw 3449  Tr wtr 3958  Ord word 4213  Oncon0 4214  suc csuc 4216 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222 This theorem is referenced by: (None)
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