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Theorem onsucmin 4178
Description: The successor of an ordinal number is the smallest larger ordinal number. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
onsucmin (A On → suc A = {x On ∣ A x})
Distinct variable group:   x,A

Proof of Theorem onsucmin
StepHypRef Expression
1 eloni 4057 . . . . 5 (x On → Ord x)
2 ordelsuc 4177 . . . . 5 ((A On Ord x) → (A x ↔ suc Ax))
31, 2sylan2 270 . . . 4 ((A On x On) → (A x ↔ suc Ax))
43rabbidva 2522 . . 3 (A On → {x On ∣ A x} = {x On ∣ suc Ax})
54inteqd 3590 . 2 (A On → {x On ∣ A x} = {x On ∣ suc Ax})
6 sucelon 4175 . . 3 (A On ↔ suc A On)
7 intmin 3605 . . 3 (suc A On → {x On ∣ suc Ax} = suc A)
86, 7sylbi 114 . 2 (A On → {x On ∣ suc Ax} = suc A)
95, 8eqtr2d 2051 1 (A On → suc A = {x On ∣ A x})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1226   wcel 1370  {crab 2284  wss 2890   cint 3585  Ord word 4044  Oncon0 4045  suc csuc 4047
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-uni 3551  df-int 3586  df-tr 3825  df-iord 4048  df-on 4050  df-suc 4053
This theorem is referenced by: (None)
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