Theorem oneqmin 3024

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 1537	. . . 4 |- (A = |^|B -> (A e. B <-> |^|B e. B))
2		onint 3012	. . . 4 |- ((B (_ On /\ B =/= (/)) -> |^|B e. B)
3	1, 2	syl5cbir 211	. . 3 |- ((B (_ On /\ B =/= (/)) -> (A = |^|B -> A e. B))
4		eleq2 1538	. . . . . . 7 |- (A = |^|B -> (x e. A <-> x e. |^|B))
5	4	biimpd 153	. . . . . 6 |- (A = |^|B -> (x e. A -> x e. |^|B))
6		onnmin 3021	. . . . . . . 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 1721	. . . 4 |- (B (_ On -> (A = |^|B -> A.x e. A -. x e. B))
11	10	adantr 391	. . 3 |- ((B (_ On /\ B =/= (/)) -> (A = |^|B -> A.x e. A -. x e. B))
12	3, 11	jcad 602	. 2 |- ((B (_ On /\ B =/= (/)) -> (A = |^|B -> (A e. B /\ A.x e. A -. x e. B)))
13		oneqmini 3023	. . 3 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
14	13	adantr 391	. 2 |- ((B (_ On /\ B =/= (/)) -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))
15	12, 14	impbid 518	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 958   e. wcel 960   =/= wne 1588  A.wral 1648   (_ wss 2050  (/)c0 2283  |^|cint 2537  Oncon0 2954

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958

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