Theorem oneqmini 3714

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 C_ 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 2615	. . . . . . . . . . . 12 |- (B C_ On -> (A e. B -> A e. On))
2		ssel 2615	. . . . . . . . . . . 12 |- (B C_ On -> (x e. B -> x e. On))
3	1, 2	anim12d 617	. . . . . . . . . . 11 |- (B C_ On -> ((A e. B /\ x e. B) -> (A e. On /\ x e. On)))
4		ontri1 3695	. . . . . . . . . . 11 |- ((A e. On /\ x e. On) -> (A C_ x <-> -. x e. A))
5	3, 4	syl6 25	. . . . . . . . . 10 |- (B C_ On -> ((A e. B /\ x e. B) -> (A C_ x <-> -. x e. A)))
6	5	expdimp 406	. . . . . . . . 9 |- ((B C_ On /\ A e. B) -> (x e. B -> (A C_ x <-> -. x e. A)))
7	6	pm5.74d 645	. . . . . . . 8 |- ((B C_ On /\ A e. B) -> ((x e. B -> A C_ x) <-> (x e. B -> -. x e. A)))
8		con2b 182	. . . . . . . 8 |- ((x e. B -> -. x e. A) <-> (x e. A -> -. x e. B))
9	7, 8	syl6bb 595	. . . . . . 7 |- ((B C_ On /\ A e. B) -> ((x e. B -> A C_ x) <-> (x e. A -> -. x e. B)))
10	9	ralbidv2 2125	. . . . . 6 |- ((B C_ On /\ A e. B) -> (A.x e. B A C_ x <-> A.x e. A -. x e. B))
11		ssint 3232	. . . . . 6 |- (A C_ |^|B <-> A.x e. B A C_ x)
12	10, 11	syl5bb 591	. . . . 5 |- ((B C_ On /\ A e. B) -> (A C_ |^|B <-> A.x e. A -. x e. B))
13	12	biimprd 171	. . . 4 |- ((B C_ On /\ A e. B) -> (A.x e. A -. x e. B -> A C_ |^|B))
14	13	expimpd 404	. . 3 |- (B C_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A C_ |^|B))
15		intss1 3231	. . . . 5 |- (A e. B -> |^|B C_ A)
16	15	a1i 8	. . . 4 |- (B C_ On -> (A e. B -> |^|B C_ A))
17	16	adantrd 427	. . 3 |- (B C_ On -> ((A e. B /\ A.x e. A -. x e. B) -> |^|B C_ A))
18	14, 17	jcad 661	. 2 |- (B C_ On -> ((A e. B /\ A.x e. A -. x e. B) -> (A C_ |^|B /\ |^|B C_ A)))
19		eqss 2631	. 2 |- (A = |^|B <-> (A C_ |^|B /\ |^|B C_ A))
20	18, 19	syl6ibr 230	1 |- (B C_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  |^|cint 3214  Oncon0 3657

This theorem is referenced by:  oneqmin 3886  alephval2 6050  alephval3 6051  cfsuc 6063

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

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