Theorem onsucmin 4629

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 4496	. . . . 5 |- ( x e. On -> Ord x )
2		ordelsuc 4627	. . . . 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 2801	. . 3 |- ( A e. On -> { x e. On | A e. x } = { x e. On | suc A C_ x } )
5	4	inteqd 3954	. 2 |- ( A e. On -> |^| { x e. On | A e. x } = |^| { x e. On | suc A C_ x } )
6		onsucb 4625	. . 3 |- ( A e. On <-> suc A e. On )
7		intmin 3969	. . 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 2266	1 |- ( A e. On -> suc A = |^| { x e. On | A e. x } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   {crab 2524   C_ wss 3211   |^|cint 3949   Ord word 4483   Oncon0 4484   suc csuc 4486

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492

This theorem is referenced by: (None)