Theorem onsucmin 2323

Description: The successor of an ordinal number is the smallest larger ordinal number.

Assertion

Ref	Expression
onsucmin	⊢ (A ∈ On → suc A = ∩{x ∈ On∣A ∈ x})

Distinct variable group(s):   x,A

Proof of Theorem onsucmin

Step	Hyp	Ref	Expression
1		sucelon 2319	. . 3 ⊢ (A ∈ On ↔ suc A ∈ On)
2		intmin 1982	. . 3 ⊢ (suc A ∈ On → suc A = ∩{x ∈ On∣suc A ⊆ x})
3	1, 2	sylbi 174	. 2 ⊢ (A ∈ On → suc A = ∩{x ∈ On∣suc A ⊆ x})
4		ordelsuc 2322	. . . . . 6 ⊢ ((A ∈ On ∧ Ord x) → (A ∈ x ↔ suc A ⊆ x))
5		eloni 2209	. . . . . 6 ⊢ (x ∈ On → Ord x)
6	4, 5	sylan2 346	. . . . 5 ⊢ ((A ∈ On ∧ x ∈ On) → (A ∈ x ↔ suc A ⊆ x))
7	6	exp 291	. . . 4 ⊢ (A ∈ On → (x ∈ On → (A ∈ x ↔ suc A ⊆ x)))
8	7	birabdv 1343	. . 3 ⊢ (A ∈ On → {x ∈ On∣A ∈ x} = {x ∈ On∣suc A ⊆ x})
9	8	inteqd 1970	. 2 ⊢ (A ∈ On → ∩{x ∈ On∣A ∈ x} = ∩{x ∈ On∣suc A ⊆ x})
10	3, 9	eqtr4d 1131	1 ⊢ (A ∈ On → suc A = ∩{x ∈ On∣A ∈ x})

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  {crab 1204   ⊆ wss 1487  ∩cint 1965  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  ranksn 3532

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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