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Theorem onsucmin 7761
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 6332 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
2 ordelsuc 7760 . . . . 5 ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴𝑥 ↔ suc 𝐴𝑥))
31, 2sylan2 594 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ suc 𝐴𝑥))
43rabbidva 3417 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
54inteqd 4917 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
6 onsucb 7757 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7 intmin 4934 . . 3 (suc 𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
86, 7sylbi 216 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
95, 8eqtr2d 2778 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 3410  wss 3915   cint 4912  Ord word 6321  Oncon0 6322  suc csuc 6324
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 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
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 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326  df-suc 6328
This theorem is referenced by:  ranksnb  9770  nadd1suc  41737  naddsuc2  41738
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