Theorem onsucmin 4649

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

Assertion

Ref	Expression
onsucmin	|- ( A e. On -> suc A = |^| { x e. On | A e. x } )

Distinct variable group:   x, A

Proof of Theorem onsucmin

Step	Hyp	Ref	Expression
1		eloni 4515	. . . . 5 |- ( x e. On -> Ord x )
2		ordelsuc 4647	. . . . 5 |- ( ( A e. On /\ Ord x ) -> ( A e. x <-> suc A C_ x ) )
3	1, 2	sylan2 286	. . . 4 |- ( ( A e. On /\ x e. On ) -> ( A e. x <-> suc A C_ x ) )
4	3	rabbidva 2809	. . 3 |- ( A e. On -> { x e. On | A e. x } = { x e. On | suc A C_ x } )
5	4	inteqd 3970	. 2 |- ( A e. On -> |^| { x e. On | A e. x } = |^| { x e. On | suc A C_ x } )
6		onsucb 4645	. . 3 |- ( A e. On <-> suc A e. On )
7		intmin 3985	. . 3 |- ( suc A e. On -> |^| { x e. On | suc A C_ x } = suc A )
8	6, 7	sylbi 121	. 2 |- ( A e. On -> |^| { x e. On | suc A C_ x } = suc A )
9	5, 8	eqtr2d 2272	1 |- ( A e. On -> suc A = |^| { x e. On | A e. x } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   |^|cint 3965   Ord word 4502   Oncon0 4503   suc csuc 4505

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511

This theorem is referenced by: (None)