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Theorem onsucmin 4198
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 4078 . . . . 5 (x On → Ord x)
2 ordelsuc 4197 . . . . 5 ((A On Ord x) → (A x ↔ suc Ax))
31, 2sylan2 270 . . . 4 ((A On x On) → (A x ↔ suc Ax))
43rabbidva 2542 . . 3 (A On → {x On ∣ A x} = {x On ∣ suc Ax})
54inteqd 3611 . 2 (A On → {x On ∣ A x} = {x On ∣ suc Ax})
6 sucelon 4195 . . 3 (A On ↔ suc A On)
7 intmin 3626 . . 3 (suc A On → {x On ∣ suc Ax} = suc A)
86, 7sylbi 114 . 2 (A On → {x On ∣ suc Ax} = suc A)
95, 8eqtr2d 2070 1 (A On → suc A = {x On ∣ A x})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  {crab 2304  wss 2911   cint 3606  Ord word 4065  Oncon0 4066  suc csuc 4068
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074
This theorem is referenced by: (None)
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