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Theorem onsucmin 7804
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 6371 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
2 ordelsuc 7803 . . . . 5 ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴𝑥 ↔ suc 𝐴𝑥))
31, 2sylan2 594 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ suc 𝐴𝑥))
43rabbidva 3440 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
54inteqd 4954 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
6 onsucb 7800 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7 intmin 4971 . . 3 (suc 𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
86, 7sylbi 216 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
95, 8eqtr2d 2774 1 (𝐴 ∈ On → suc 𝐴 = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {crab 3433  wss 3947   cint 4949  Ord word 6360  Oncon0 6361  suc csuc 6363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-suc 6367
This theorem is referenced by:  ranksnb  9818  nadd1suc  42075  naddsuc2  42076
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