Theorem onsucmin 7643

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 6261	. . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥)
2		ordelsuc 7642	. . . . 5 ⊢ ((𝐴 ∈ On ∧ Ord 𝑥) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
3	1, 2	sylan2 592	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ∈ 𝑥 ↔ suc 𝐴 ⊆ 𝑥))
4	3	rabbidva 3402	. . 3 ⊢ (𝐴 ∈ On → {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥})
5	4	inteqd 4881	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥} = ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥})
6		sucelon 7639	. . 3 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
7		intmin 4896	. . 3 ⊢ (suc 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴)
8	6, 7	sylbi 216	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ suc 𝐴 ⊆ 𝑥} = suc 𝐴)
9	5, 8	eqtr2d 2779	1 ⊢ (𝐴 ∈ On → suc 𝐴 = ∩ {𝑥 ∈ On ∣ 𝐴 ∈ 𝑥})

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {crab 3067   ⊆ wss 3883  ∩ cint 4876  Ord word 6250  Oncon0 6251  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-suc 6257

This theorem is referenced by:  ranksnb  9516