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Theorem onsucmin 4261
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 ∣ 𝐴𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem onsucmin
StepHypRef Expression
1 eloni 4140 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
2 ordelsuc 4259 . . . . 5 ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴𝑥 ↔ suc 𝐴𝑥))
31, 2sylan2 274 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ suc 𝐴𝑥))
43rabbidva 2565 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
54inteqd 3648 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
6 sucelon 4257 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7 intmin 3663 . . 3 (suc 𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
86, 7sylbi 118 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
95, 8eqtr2d 2089 1 (𝐴 ∈ On → suc 𝐴 = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  {crab 2327  wss 2945   cint 3643  Ord word 4127  Oncon0 4128  suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by: (None)
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