Theorem oneqmin 3008

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 (_ 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 1526	. . . 4 |- (A = |^|B -> (A e. B <-> |^|B e. B))
2		onint 2996	. . . 4 |- ((B (_ On /\ B =/= (/)) -> |^|B e. B)
3	1, 2	syl5cbir 211	. . 3 |- ((B (_ On /\ B =/= (/)) -> (A = |^|B -> A e. B))
4		eleq2 1527	. . . . . . 7 |- (A = |^|B -> (x e. A <-> x e. |^|B))
5	4	biimpd 153	. . . . . 6 |- (A = |^|B -> (x e. A -> x e. |^|B))
6		onnmin 3005	. . . . . . . 8 |- ((B (_ On /\ x e. B) -> -. x e. |^|B)
7	6	ex 373	. . . . . . 7 |- (B (_ On -> (x e. B -> -. x e. |^|B))
8	7	con2d 91	. . . . . 6 |- (B (_ On -> (x e. |^|B -> -. x e. B))
9	5, 8	syl9r 58	. . . . 5 |- (B (_ On -> (A = |^|B -> (x e. A -> -. x e. B)))
10	9	r19.21adv 1710	. . . 4 |- (B (_ On -> (A = |^|B -> A.x e. A -. x e. B))
11	10	adantr 389	. . 3 |- ((B (_ On /\ B =/= (/)) -> (A = |^|B -> A.x e. A -. x e. B))
12	3, 11	jcad 598	. 2 |- ((B (_ On /\ B =/= (/)) -> (A = |^|B -> (A e. B /\ A.x e. A -. x e. B)))
13		oneqmini 3007	. . 3 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
14	13	adantr 389	. 2 |- ((B (_ On /\ B =/= (/)) -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
15	12, 14	impbid 514	1 |- ((B (_ On /\ B =/= (/)) -> (A = |^|B <-> (A e. B /\ A.x e. A -. x e. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637   (_ wss 2037  (/)c0 2270  |^|cint 2523  Oncon0 2938

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  ax-un 2857

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-tp 2405  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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