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Theorem ordpwsucss 4578
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 4383 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both suc 𝐴 = 𝐴 (onunisuci 4444) and {𝑥 ∈ On ∣ 𝑥𝐴} = 𝐴 (onuniss2 4523).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4581). (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 4574 . . . . 5 (Ord 𝐴 ↔ Ord suc 𝐴)
2 ordelon 4395 . . . . . 6 ((Ord suc 𝐴𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
32ex 115 . . . . 5 (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
41, 3sylbi 121 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ On))
5 ordtr 4390 . . . . 5 (Ord 𝐴 → Tr 𝐴)
6 trsucss 4435 . . . . 5 (Tr 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
75, 6syl 14 . . . 4 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥𝐴))
84, 7jcad 307 . . 3 (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥𝐴)))
9 elin 3330 . . . 4 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴𝑥 ∈ On))
10 velpw 3594 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1110anbi2ci 459 . . . 4 ((𝑥 ∈ 𝒫 𝐴𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
129, 11bitri 184 . . 3 (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥𝐴))
138, 12imbitrrdi 162 . 2 (Ord 𝐴 → (𝑥 ∈ suc 𝐴𝑥 ∈ (𝒫 𝐴 ∩ On)))
1413ssrdv 3173 1 (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2158  cin 3140  wss 3141  𝒫 cpw 3587  Tr wtr 4113  Ord word 4374  Oncon0 4375  suc csuc 4377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-uni 3822  df-tr 4114  df-iord 4378  df-on 4380  df-suc 4383
This theorem is referenced by: (None)
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