Theorem onsucmin 4233

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 4112	. . . . 5 |- ( x e. On -> Ord x )
2		ordelsuc 4231	. . . . 5 |- ( ( A e. On /\ Ord x ) -> ( A e. x <-> suc A C_ x ) )
3	1, 2	sylan2 270	. . . 4 |- ( ( A e. On /\ x e. On ) -> ( A e. x <-> suc A C_ x ) )
4	3	rabbidva 2548	. . 3 |- ( A e. On -> { x e. On | A e. x } = { x e. On | suc A C_ x } )
5	4	inteqd 3620	. 2 |- ( A e. On -> |^| { x e. On | A e. x } = |^| { x e. On | suc A C_ x } )
6		sucelon 4229	. . 3 |- ( A e. On <-> suc A e. On )
7		intmin 3635	. . 3 |- ( suc A e. On -> |^| { x e. On | suc A C_ x } = suc A )
8	6, 7	sylbi 114	. 2 |- ( A e. On -> |^| { x e. On | suc A C_ x } = suc A )
9	5, 8	eqtr2d 2073	1 |- ( A e. On -> suc A = |^| { x e. On | A e. x } )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   {crab 2310   C_ wss 2917   |^|cint 3615   Ord word 4099   Oncon0 4100   suc csuc 4102

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by: (None)