Theorem oneqmini 3007

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 e. B /\ A.x e. A -. x e. B) -> A = |^|B))

Distinct variable groups:   x,A   x,B

Proof of Theorem oneqmini

Step	Hyp	Ref	Expression
1		ssel 2053	. . . . . . . . . . . . . 14 |- (B (_ On -> (A e. B -> A e. On))
2		ssel 2053	. . . . . . . . . . . . . 14 |- (B (_ On -> (x e. B -> x e. On))
3	1, 2	anim12d 556	. . . . . . . . . . . . 13 |- (B (_ On -> ((A e. B /\ x e. B) -> (A e. On /\ x e. On)))
4		ontri1 2971	. . . . . . . . . . . . 13 |- ((A e. On /\ x e. On) -> (A (_ x <-> -. x e. A))
5	3, 4	syl6 22	. . . . . . . . . . . 12 |- (B (_ On -> ((A e. B /\ x e. B) -> (A (_ x <-> -. x e. A)))
6	5	exp3a 375	. . . . . . . . . . 11 |- (B (_ On -> (A e. B -> (x e. B -> (A (_ x <-> -. x e. A))))
7	6	imp 350	. . . . . . . . . 10 |- ((B (_ On /\ A e. B) -> (x e. B -> (A (_ x <-> -. x e. A)))
8	7	pm5.74d 583	. . . . . . . . 9 |- ((B (_ On /\ A e. B) -> ((x e. B -> A (_ x) <-> (x e. B -> -. x e. A)))
9		bi2.03 165	. . . . . . . . 9 |- ((x e. B -> -. x e. A) <-> (x e. A -> -. x e. B))
10	8, 9	syl6bb 534	. . . . . . . 8 |- ((B (_ On /\ A e. B) -> ((x e. B -> A (_ x) <-> (x e. A -> -. x e. B)))
11	10	ralbidv2 1657	. . . . . . 7 |- ((B (_ On /\ A e. B) -> (A.x e. B A (_ x <-> A.x e. A -. x e. B))
12		ssint 2539	. . . . . . 7 |- (A (_ |^|B <-> A.x e. B A (_ x)
13	11, 12	syl5bb 530	. . . . . 6 |- ((B (_ On /\ A e. B) -> (A (_ |^|B <-> A.x e. A -. x e. B))
14	13	biimprd 154	. . . . 5 |- ((B (_ On /\ A e. B) -> (A.x e. A -. x e. B -> A (_ |^|B))
15	14	ex 373	. . . 4 |- (B (_ On -> (A e. B -> (A.x e. A -. x e. B -> A (_ |^|B)))
16	15	imp3a 361	. . 3 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A (_ |^|B))
17		intss1 2538	. . . . 5 |- (A e. B -> |^|B (_ A)
18	17	a1i 8	. . . 4 |- (B (_ On -> (A e. B -> |^|B (_ A))
19	18	adantrd 391	. . 3 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> |^|B (_ A))
20	16, 19	jcad 598	. 2 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> (A (_ |^|B /\ |^|B (_ A)))
21		eqss 2067	. 2 |- (A = |^|B <-> (A (_ |^|B /\ |^|B (_ A))
22	20, 21	syl6ibr 213	1 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637   (_ wss 2037  |^|cint 2523  Oncon0 2938

This theorem is referenced by:  oneqmin 3008  alephval2 4874  alephval3 4875  cfsuc 4887

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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