Theorem onsucmin 4198

Description: The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)

Assertion

Ref	Expression
onsucmin	⊢ (A ∈ On → suc A = ∩ {x ∈ On ∣ A ∈ x})

Distinct variable group:   x,A

Proof of Theorem onsucmin

Step	Hyp	Ref	Expression
1		eloni 4078	. . . . 5 ⊢ (x ∈ On → Ord x)
2		ordelsuc 4197	. . . . 5 ⊢ ((A ∈ On ∧ Ord x) → (A ∈ x ↔ suc A ⊆ x))
3	1, 2	sylan2 270	. . . 4 ⊢ ((A ∈ On ∧ x ∈ On) → (A ∈ x ↔ suc A ⊆ x))
4	3	rabbidva 2542	. . 3 ⊢ (A ∈ On → {x ∈ On ∣ A ∈ x} = {x ∈ On ∣ suc A ⊆ x})
5	4	inteqd 3611	. 2 ⊢ (A ∈ On → ∩ {x ∈ On ∣ A ∈ x} = ∩ {x ∈ On ∣ suc A ⊆ x})
6		sucelon 4195	. . 3 ⊢ (A ∈ On ↔ suc A ∈ On)
7		intmin 3626	. . 3 ⊢ (suc A ∈ On → ∩ {x ∈ On ∣ suc A ⊆ x} = suc A)
8	6, 7	sylbi 114	. 2 ⊢ (A ∈ On → ∩ {x ∈ On ∣ suc A ⊆ x} = suc A)
9	5, 8	eqtr2d 2070	1 ⊢ (A ∈ On → suc A = ∩ {x ∈ On ∣ A ∈ x})

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {crab 2304   ⊆ wss 2911  ∩ cint 3606  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-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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

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-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074

This theorem is referenced by: (None)