Theorem oneqmin 3886

Description: A way to show that an ordinal number equals the minimum of a non-empty 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
oneqmin	|- ((B C_ On /\ B =/= (/)) -> (A = |^|B <-> (A e. B /\ A.x e. A -. x e. B)))

Distinct variable groups:   x,A   x,B

Proof of Theorem oneqmin

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . 4 |- (A = |^|B -> (A e. B <-> |^|B e. B))
2		onint 3876	. . . 4 |- ((B C_ On /\ B =/= (/)) -> |^|B e. B)
3	1, 2	syl5cbir 228	. . 3 |- ((B C_ On /\ B =/= (/)) -> (A = |^|B -> A e. B))
4		eleq2 1958	. . . . . . 7 |- (A = |^|B -> (x e. A <-> x e. |^|B))
5	4	biimpd 170	. . . . . 6 |- (A = |^|B -> (x e. A -> x e. |^|B))
6		onnmin 3884	. . . . . . . 8 |- ((B C_ On /\ x e. B) -> -. x e. |^|B)
7	6	ex 402	. . . . . . 7 |- (B C_ On -> (x e. B -> -. x e. |^|B))
8	7	con2d 107	. . . . . 6 |- (B C_ On -> (x e. |^|B -> -. x e. B))
9	5, 8	syl9r 72	. . . . 5 |- (B C_ On -> (A = |^|B -> (x e. A -> -. x e. B)))
10	9	r19.21adv 2181	. . . 4 |- (B C_ On -> (A = |^|B -> A.x e. A -. x e. B))
11	10	adantr 425	. . 3 |- ((B C_ On /\ B =/= (/)) -> (A = |^|B -> A.x e. A -. x e. B))
12	3, 11	jcad 661	. 2 |- ((B C_ On /\ B =/= (/)) -> (A = |^|B -> (A e. B /\ A.x e. A -. x e. B)))
13		oneqmini 3714	. . 3 |- (B C_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
14	13	adantr 425	. 2 |- ((B C_ On /\ B =/= (/)) -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
15	12, 14	impbid 574	1 |- ((B C_ On /\ B =/= (/)) -> (A = |^|B <-> (A e. B /\ A.x e. A -. x e. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105   C_ wss 2593  (/)c0 2875  |^|cint 3214  Oncon0 3657

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-13 1311  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  ax-un 3790

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-tp 3052  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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