Theorem ordpwsucss 4243

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 4074 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if A ∈ On then both ∪ suc A = A (onunisuci 4135) and ∪ {x ∈ On ∣ x ⊆ A} = A (onuniss2 4203).

Constructively (𝒫 A ∩ On) and suc A cannot be shown to be equivalent (as proved at ordpwsucexmid 4246). (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 4241	. . . . 5 ⊢ (Ord A ↔ Ord suc A)
2		ordelon 4086	. . . . . 6 ⊢ ((Ord suc A ∧ x ∈ suc A) → x ∈ On)
3	2	ex 108	. . . . 5 ⊢ (Ord suc A → (x ∈ suc A → x ∈ On))
4	1, 3	sylbi 114	. . . 4 ⊢ (Ord A → (x ∈ suc A → x ∈ On))
5		ordtr 4081	. . . . 5 ⊢ (Ord A → Tr A)
6		trsucss 4126	. . . . 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 291	. . 3 ⊢ (Ord A → (x ∈ suc A → (x ∈ On ∧ x ⊆ A)))
9		elin 3120	. . . 4 ⊢ (x ∈ (𝒫 A ∩ On) ↔ (x ∈ 𝒫 A ∧ x ∈ On))
10		selpw 3358	. . . . 5 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A)
11	10	anbi2ci 432	. . . 4 ⊢ ((x ∈ 𝒫 A ∧ x ∈ On) ↔ (x ∈ On ∧ x ⊆ A))
12	9, 11	bitri 173	. . 3 ⊢ (x ∈ (𝒫 A ∩ On) ↔ (x ∈ On ∧ x ⊆ A))
13	8, 12	syl6ibr 151	. 2 ⊢ (Ord A → (x ∈ suc A → x ∈ (𝒫 A ∩ On)))
14	13	ssrdv 2945	1 ⊢ (Ord A → suc A ⊆ (𝒫 A ∩ On))

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390   ∩ cin 2910   ⊆ wss 2911  𝒫 cpw 3351  Tr wtr 3845  Ord word 4065  Oncon0 4066  suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074

This theorem is referenced by: (None)