Theorem onpwsuc 2315

Description: The collection of ordinal numbers in the power set of an ordinal number is its successor.

Assertion

Ref	Expression
onpwsuc	⊢ (A ∈ On → (℘A ∩ On) = suc A)

Proof of Theorem onpwsuc

Step	Hyp	Ref	Expression
1		onsssuc 2311	. . . . . . 7 ⊢ ((x ∈ On ∧ A ∈ On) → (x ⊆ A ↔ x ∈ suc A))
2	1	exp 291	. . . . . 6 ⊢ (x ∈ On → (A ∈ On → (x ⊆ A ↔ x ∈ suc A)))
3	2	com12 13	. . . . 5 ⊢ (A ∈ On → (x ∈ On → (x ⊆ A ↔ x ∈ suc A)))
4	3	pm5.32d 491	. . . 4 ⊢ (A ∈ On → ((x ∈ On ∧ x ⊆ A) ↔ (x ∈ On ∧ x ∈ suc A)))
5		pm3.27 260	. . . . . 6 ⊢ ((x ∈ On ∧ x ∈ suc A) → x ∈ suc A)
6	5	a1i 7	. . . . 5 ⊢ (A ∈ On → ((x ∈ On ∧ x ∈ suc A) → x ∈ suc A))
7		suceloni 2314	. . . . . . 7 ⊢ (A ∈ On → suc A ∈ On)
8		onelon 2223	. . . . . . . 8 ⊢ ((suc A ∈ On ∧ x ∈ suc A) → x ∈ On)
9	8	exp 291	. . . . . . 7 ⊢ (suc A ∈ On → (x ∈ suc A → x ∈ On))
10	7, 9	syl 12	. . . . . 6 ⊢ (A ∈ On → (x ∈ suc A → x ∈ On))
11	10	ancrd 247	. . . . 5 ⊢ (A ∈ On → (x ∈ suc A → (x ∈ On ∧ x ∈ suc A)))
12	6, 11	impbid 397	. . . 4 ⊢ (A ∈ On → ((x ∈ On ∧ x ∈ suc A) ↔ x ∈ suc A))
13	4, 12	bitrd 406	. . 3 ⊢ (A ∈ On → ((x ∈ On ∧ x ⊆ A) ↔ x ∈ suc A))
14		elin 1635	. . . 4 ⊢ (x ∈ (℘A ∩ On) ↔ (x ∈ ℘A ∧ x ∈ On))
15		visset 1350	. . . . . 6 ⊢ x ∈ V
16	15	elpw 1801	. . . . 5 ⊢ (x ∈ ℘A ↔ x ⊆ A)
17	16	anbi1i 368	. . . 4 ⊢ ((x ∈ ℘A ∧ x ∈ On) ↔ (x ⊆ A ∧ x ∈ On))
18		ancom 333	. . . 4 ⊢ ((x ⊆ A ∧ x ∈ On) ↔ (x ∈ On ∧ x ⊆ A))
19	14, 17, 18	3bitr 155	. . 3 ⊢ (x ∈ (℘A ∩ On) ↔ (x ∈ On ∧ x ⊆ A))
20	13, 19	syl5bb 410	. 2 ⊢ (A ∈ On → (x ∈ (℘A ∩ On) ↔ x ∈ suc A))
21	20	cleqrd 1100	1 ⊢ (A ∈ On → (℘A ∩ On) = suc A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487  ℘cpw 1798  Oncon0 2199  suc csuc 2201

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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