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Theorem onsucmin 7530
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 6188 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
2 ordelsuc 7529 . . . . 5 ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴𝑥 ↔ suc 𝐴𝑥))
31, 2sylan2 595 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ suc 𝐴𝑥))
43rabbidva 3463 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
54inteqd 4867 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
6 sucelon 7526 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7 intmin 4882 . . 3 (suc 𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
86, 7sylbi 220 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
95, 8eqtr2d 2860 1 (𝐴 ∈ On → suc 𝐴 = {𝑥 ∈ On ∣ 𝐴𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  {crab 3137  wss 3919   cint 4862  Ord word 6177  Oncon0 6178  suc csuc 6180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182  df-suc 6184
This theorem is referenced by:  ranksnb  9253
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