Theorem ordpwsucss 4568

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 4373 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4434) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4513).

Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4571). (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 4564	. . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
2		ordelon 4385	. . . . . 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 4380	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
6		trsucss 4425	. . . . 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 3320	. . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On))
10		velpw 3584	. . . . 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 3163	1 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2148   ∩ cin 3130   ⊆ wss 3131  𝒫 cpw 3577  Tr wtr 4103  Ord word 4364  Oncon0 4365  suc csuc 4367

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-uni 3812  df-tr 4104  df-iord 4368  df-on 4370  df-suc 4373

This theorem is referenced by: (None)