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Theorem onsucmin 7063
 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 5771 . . . . 5 (𝑥 ∈ On → Ord 𝑥)
2 ordelsuc 7062 . . . . 5 ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴𝑥 ↔ suc 𝐴𝑥))
31, 2sylan2 490 . . . 4 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴𝑥 ↔ suc 𝐴𝑥))
43rabbidva 3219 . . 3 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
54inteqd 4512 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴𝑥} = {𝑥 ∈ On ∣ suc 𝐴𝑥})
6 sucelon 7059 . . 3 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7 intmin 4529 . . 3 (suc 𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
86, 7sylbi 207 . 2 (𝐴 ∈ On → {𝑥 ∈ On ∣ suc 𝐴𝑥} = suc 𝐴)
95, 8eqtr2d 2686 1 (𝐴 ∈ On → suc 𝐴 = {𝑥 ∈ On ∣ 𝐴𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1523   ∈ wcel 2030  {crab 2945   ⊆ wss 3607  ∩ cint 4507  Ord word 5760  Oncon0 5761  suc csuc 5763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767 This theorem is referenced by:  ranksnb  8728
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