Theorem oneqmin 2273

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 group(s):   x,A   x,B

Proof of Theorem oneqmin

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 |- (A = |^|B -> (A e. B <-> |^|B e. B))
2		onint 2261	. . . . 5 |- ((B (_ On /\ -. B = (/)) -> |^|B e. B)
3	1, 2	syl5bir 184	. . . 4 |- (A = |^|B -> ((B (_ On /\ -. B = (/)) -> A e. B))
4	3	com12 13	. . 3 |- ((B (_ On /\ -. B = (/)) -> (A = |^|B -> A e. B))
5		eleq2 1150	. . . . . . 7 |- (A = |^|B -> (x e. A <-> x e. |^|B))
6	5	biimpd 135	. . . . . 6 |- (A = |^|B -> (x e. A -> x e. |^|B))
7		onnmin 2270	. . . . . . . 8 |- ((B (_ On /\ x e. B) -> -. x e. |^|B)
8	7	exp 291	. . . . . . 7 |- (B (_ On -> (x e. B -> -. x e. |^|B))
9	8	con2d 83	. . . . . 6 |- (B (_ On -> (x e. |^|B -> -. x e. B))
10	6, 9	syl9r 56	. . . . 5 |- (B (_ On -> (A = |^|B -> (x e. A -> -. x e. B)))
11	10	r19.21adv 1262	. . . 4 |- (B (_ On -> (A = |^|B -> A.x e. A -. x e. B))
12	11	adantr 306	. . 3 |- ((B (_ On /\ -. B = (/)) -> (A = |^|B -> A.x e. A -. x e. B))
13	4, 12	jcad 455	. 2 |- ((B (_ On /\ -. B = (/)) -> (A = |^|B -> (A e. B /\ A.x e. A -. x e. B)))
14		oneqmini 2272	. . 3 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
15	14	adantr 306	. 2 |- ((B (_ On /\ -. B = (/)) -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
16	13, 15	impbid 397	1 |- ((B (_ On /\ -. B = (/)) -> (A = |^|B <-> (A e. B /\ A.x e. A -. x e. B)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201   (_ wss 1487  (/)c0 1707  |^|cint 1965  Oncon0 2199

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-un 1076  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-tp 1814  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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