Theorem onsucmin 4491

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 4360	. . . . 5 |- ( x e. On -> Ord x )
2		ordelsuc 4489	. . . . 5 |- ( ( A e. On /\ Ord x ) -> ( A e. x <-> suc A C_ x ) )
3	1, 2	sylan2 284	. . . 4 |- ( ( A e. On /\ x e. On ) -> ( A e. x <-> suc A C_ x ) )
4	3	rabbidva 2718	. . 3 |- ( A e. On -> { x e. On | A e. x } = { x e. On | suc A C_ x } )
5	4	inteqd 3836	. 2 |- ( A e. On -> |^| { x e. On | A e. x } = |^| { x e. On | suc A C_ x } )
6		sucelon 4487	. . 3 |- ( A e. On <-> suc A e. On )
7		intmin 3851	. . 3 |- ( suc A e. On -> |^| { x e. On | suc A C_ x } = suc A )
8	6, 7	sylbi 120	. 2 |- ( A e. On -> |^| { x e. On | suc A C_ x } = suc A )
9	5, 8	eqtr2d 2204	1 |- ( A e. On -> suc A = |^| { x e. On | A e. x } )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   {crab 2452   C_ wss 3121   |^|cint 3831   Ord word 4347   Oncon0 4348   suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356

This theorem is referenced by: (None)