Theorem ordpwsucss 4599

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 4402 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4463) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4544).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4602). (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.

Step	Hyp	Ref	Expression
1		ordsuc 4595	. . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
2		ordelon 4414	. . . . . 6 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
3	2	ex 115	. . . . 5 ⊢ (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
4	1, 3	sylbi 121	. . . 4 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
5		ordtr 4409	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
6		trsucss 4454	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴))
7	5, 6	syl 14	. . . 4 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴))
8	4, 7	jcad 307	. . 3 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)))
9		elin 3342	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On))
10		velpw 3608	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
11	10	anbi2ci 459	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
12	9, 11	bitri 184	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
13	8, 12	imbitrrdi 162	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ (𝒫 𝐴 ∩ On)))
14	13	ssrdv 3185	1 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2164   ∩ cin 3152   ⊆ wss 3153  𝒫 cpw 3601  Tr wtr 4127  Ord word 4393  Oncon0 4394  suc csuc 4396

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402

This theorem is referenced by: (None)