Theorem onsucmin 4491

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 4360	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
2		ordelsuc 4489	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
3	1, 2	sylan2 284	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
4	3	rabbidva 2718	. . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥})
5	4	inteqd 3836	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥})
6		sucelon 4487	. . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7		intmin 3851	. . 3 ⊢ (suc 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴)
8	6, 7	sylbi 120	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴)
9	5, 8	eqtr2d 2204	1 ⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥})

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348   ∈ wcel 2141  {crab 2452   ⊆ wss 3121  ∩ cint 3831  Ord word 4347  Oncon0 4348  suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356

This theorem is referenced by: (None)