Theorem onsucmin 4178

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 4057	. . . . 5 ⊢ (x ∈ On → Ord x)
2		ordelsuc 4177	. . . . 5 ⊢ ((A ∈ On ∧ Ord x) → (A ∈ x ↔ suc A ⊆ x))
3	1, 2	sylan2 270	. . . 4 ⊢ ((A ∈ On ∧ x ∈ On) → (A ∈ x ↔ suc A ⊆ x))
4	3	rabbidva 2522	. . 3 ⊢ (A ∈ On → {x ∈ On ∣ A ∈ x} = {x ∈ On ∣ suc A ⊆ x})
5	4	inteqd 3590	. 2 ⊢ (A ∈ On → ∩ {x ∈ On ∣ A ∈ x} = ∩ {x ∈ On ∣ suc A ⊆ x})
6		sucelon 4175	. . 3 ⊢ (A ∈ On ↔ suc A ∈ On)
7		intmin 3605	. . 3 ⊢ (suc A ∈ On → ∩ {x ∈ On ∣ suc A ⊆ x} = suc A)
8	6, 7	sylbi 114	. 2 ⊢ (A ∈ On → ∩ {x ∈ On ∣ suc A ⊆ x} = suc A)
9	5, 8	eqtr2d 2051	1 ⊢ (A ∈ On → suc A = ∩ {x ∈ On ∣ A ∈ x})

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1226   ∈ wcel 1370  {crab 2284   ⊆ wss 2890  ∩ cint 3585  Ord word 4044  Oncon0 4045  suc csuc 4047

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-uni 3551  df-int 3586  df-tr 3825  df-iord 4048  df-on 4050  df-suc 4053

This theorem is referenced by: (None)