Theorem ordpwsucss 4544

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 4349 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4410) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4489).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4547). (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 4540	. . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
2		ordelon 4361	. . . . . 6 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ On)
3	2	ex 114	. . . . 5 ⊢ (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
4	1, 3	sylbi 120	. . . 4 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On))
5		ordtr 4356	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
6		trsucss 4401	. . . . 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 305	. . 3 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)))
9		elin 3305	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On))
10		velpw 3566	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
11	10	anbi2ci 455	. . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
12	9, 11	bitri 183	. . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))
13	8, 12	syl6ibr 161	. 2 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ (𝒫 𝐴 ∩ On)))
14	13	ssrdv 3148	1 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2136   ∩ cin 3115   ⊆ wss 3116  𝒫 cpw 3559  Tr wtr 4080  Ord word 4340  Oncon0 4341  suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by: (None)