Theorem onsucmin 3062

Description: The successor of an ordinal number is the smallest larger ordinal number.

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		ordelsuc 3061	. . . . 5 |- ((A e. On /\ Ord x) -> (A e. x <-> suc A (_ x))
2		eloni 2948	. . . . 5 |- (x e. On -> Ord x)
3	1, 2	sylan2 451	. . . 4 |- ((A e. On /\ x e. On) -> (A e. x <-> suc A (_ x))
4	3	rabbidv 1797	. . 3 |- (A e. On -> {x e. On | A e. x} = {x e. On | suc A (_ x})
5	4	inteqd 2528	. 2 |- (A e. On -> |^|{x e. On | A e. x} = |^|{x e. On | suc A (_ x})
6		sucelon 3058	. . 3 |- (A e. On <-> suc A e. On)
7		intmin 2543	. . 3 |- (suc A e. On -> |^|{x e. On | suc A (_ x} = suc A)
8	6, 7	sylbi 199	. 2 |- (A e. On -> |^|{x e. On | suc A (_ x} = suc A)
9	5, 8	eqtr2d 1500	1 |- (A e. On -> suc A = |^|{x e. On | A e. x})

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  {crab 1640   (_ wss 2037  |^|cint 2523  Ord word 2937  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  ranksn 4661

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

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