Theorem onsucmin 7836

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 ∣ 𝐴 ∈ 𝑥})

Distinct variable group:   𝑥,𝐴

Proof of Theorem onsucmin

Step	Hyp	Ref	Expression
1		eloni 6375	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
2		ordelsuc 7835	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
3	1, 2	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
4	3	rabbidva 3418	. . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥})
5	4	inteqd 4911	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥})
6		onsucb 7832	. . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7		intmin 4927	. . 3 ⊢ (suc 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴)
8	6, 7	sylbi 220	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴)
9	5, 8	eqtr2d 2796	1 ⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3896  ∩ cint 4906  Ord word 6364  Oncon0 6365  suc csuc 6367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6368  df-on 6369  df-suc 6371

This theorem is used by:  naddsuc2  8718  ranksnb  9844  nadd1suc  44503