Theorem oneqmini 2272

Description: A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection.

Assertion

Ref	Expression
oneqmini	⊢ (B ⊆ On → ((A ∈ B ∧ ∀x ∈ A ¬ x ∈ B) → A = ∩B))

Distinct variable group(s):   x,A   x,B

Proof of Theorem oneqmini

Step	Hyp	Ref	Expression
1		ssel 1502	. . . . . . . . . . . . . 14 ⊢ (B ⊆ On → (A ∈ B → A ∈ On))
2		ssel 1502	. . . . . . . . . . . . . 14 ⊢ (B ⊆ On → (x ∈ B → x ∈ On))
3	1, 2	anim12d 431	. . . . . . . . . . . . 13 ⊢ (B ⊆ On → ((A ∈ B ∧ x ∈ B) → (A ∈ On ∧ x ∈ On)))
4		ontri1 2232	. . . . . . . . . . . . 13 ⊢ ((A ∈ On ∧ x ∈ On) → (A ⊆ x ↔ ¬ x ∈ A))
5	3, 4	syl6 23	. . . . . . . . . . . 12 ⊢ (B ⊆ On → ((A ∈ B ∧ x ∈ B) → (A ⊆ x ↔ ¬ x ∈ A)))
6	5	exp3a 292	. . . . . . . . . . 11 ⊢ (B ⊆ On → (A ∈ B → (x ∈ B → (A ⊆ x ↔ ¬ x ∈ A))))
7	6	imp 277	. . . . . . . . . 10 ⊢ ((B ⊆ On ∧ A ∈ B) → (x ∈ B → (A ⊆ x ↔ ¬ x ∈ A)))
8	7	pm5.74d 444	. . . . . . . . 9 ⊢ ((B ⊆ On ∧ A ∈ B) → ((x ∈ B → A ⊆ x) ↔ (x ∈ B → ¬ x ∈ A)))
9		bi2.03 144	. . . . . . . . 9 ⊢ ((x ∈ B → ¬ x ∈ A) ↔ (x ∈ A → ¬ x ∈ B))
10	8, 9	syl6bb 414	. . . . . . . 8 ⊢ ((B ⊆ On ∧ A ∈ B) → ((x ∈ B → A ⊆ x) ↔ (x ∈ A → ¬ x ∈ B)))
11	10	biraldv2 1221	. . . . . . 7 ⊢ ((B ⊆ On ∧ A ∈ B) → (∀x ∈ B A ⊆ x ↔ ∀x ∈ A ¬ x ∈ B))
12		ssint 1980	. . . . . . 7 ⊢ (A ⊆ ∩B ↔ ∀x ∈ B A ⊆ x)
13	11, 12	syl5bb 410	. . . . . 6 ⊢ ((B ⊆ On ∧ A ∈ B) → (A ⊆ ∩B ↔ ∀x ∈ A ¬ x ∈ B))
14	13	biimprd 136	. . . . 5 ⊢ ((B ⊆ On ∧ A ∈ B) → (∀x ∈ A ¬ x ∈ B → A ⊆ ∩B))
15	14	exp 291	. . . 4 ⊢ (B ⊆ On → (A ∈ B → (∀x ∈ A ¬ x ∈ B → A ⊆ ∩B)))
16	15	imp3a 279	. . 3 ⊢ (B ⊆ On → ((A ∈ B ∧ ∀x ∈ A ¬ x ∈ B) → A ⊆ ∩B))
17		intss1 1979	. . . . 5 ⊢ (A ∈ B → ∩B ⊆ A)
18	17	a1i 7	. . . 4 ⊢ (B ⊆ On → (A ∈ B → ∩B ⊆ A))
19	18	adantrd 308	. . 3 ⊢ (B ⊆ On → ((A ∈ B ∧ ∀x ∈ A ¬ x ∈ B) → ∩B ⊆ A))
20	16, 19	jcad 455	. 2 ⊢ (B ⊆ On → ((A ∈ B ∧ ∀x ∈ A ¬ x ∈ B) → (A ⊆ ∩B ∧ ∩B ⊆ A)))
21		eqss 1516	. 2 ⊢ (A = ∩B ↔ (A ⊆ ∩B ∧ ∩B ⊆ A))
22	20, 21	syl6ibr 186	1 ⊢ (B ⊆ On → ((A ∈ B ∧ ∀x ∈ A ¬ x ∈ B) → A = ∩B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  ∩cint 1965  Oncon0 2199

This theorem is referenced by:  oneqmin 2273  cfsuc 3709

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-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-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-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

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