Theorem onsucmin 4629

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 4496	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
2		ordelsuc 4627	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
3	1, 2	sylan2 286	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
4	3	rabbidva 2801	. . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥})
5	4	inteqd 3954	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥})
6		onsucb 4625	. . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7		intmin 3969	. . 3 ⊢ (suc 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴)
8	6, 7	sylbi 121	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴)
9	5, 8	eqtr2d 2266	1 ⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {crab 2524   ⊆ wss 3211  ∩ cint 3949  Ord word 4483  Oncon0 4484  suc csuc 4486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-int 3950  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492

This theorem is referenced by: (None)