Theorem onsucmin 4543

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 4410	. . . . 5 |- ( x e. On -> Ord x )
2		ordelsuc 4541	. . . . 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 2751	. . 3 |- ( A e. On -> { x e. On | A e. x } = { x e. On | suc A C_ x } )
5	4	inteqd 3879	. 2 |- ( A e. On -> |^| { x e. On | A e. x } = |^| { x e. On | suc A C_ x } )
6		onsucb 4539	. . 3 |- ( A e. On <-> suc A e. On )
7		intmin 3894	. . 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 2230	1 |- ( A e. On -> suc A = |^| { x e. On | A e. x } )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   {crab 2479   C_ wss 3157   |^|cint 3874   Ord word 4397   Oncon0 4398   suc csuc 4400

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406

This theorem is referenced by: (None)