Theorem oneqmini 3017

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 (_ 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 2063	. . . . . . . . . . . . . 14 |- (B (_ On -> (A e. B -> A e. On))
2		ssel 2063	. . . . . . . . . . . . . 14 |- (B (_ On -> (x e. B -> x e. On))
3	1, 2	anim12d 558	. . . . . . . . . . . . 13 |- (B (_ On -> ((A e. B /\ x e. B) -> (A e. On /\ x e. On)))
4		ontri1 2981	. . . . . . . . . . . . 13 |- ((A e. On /\ x e. On) -> (A (_ x <-> -. x e. A))
5	3, 4	syl6 22	. . . . . . . . . . . 12 |- (B (_ On -> ((A e. B /\ x e. B) -> (A (_ x <-> -. x e. A)))
6	5	exp3a 375	. . . . . . . . . . 11 |- (B (_ On -> (A e. B -> (x e. B -> (A (_ x <-> -. x e. A))))
7	6	imp 350	. . . . . . . . . 10 |- ((B (_ On /\ A e. B) -> (x e. B -> (A (_ x <-> -. x e. A)))
8	7	pm5.74d 585	. . . . . . . . 9 |- ((B (_ On /\ A e. B) -> ((x e. B -> A (_ x) <-> (x e. B -> -. x e. A)))
9		bi2.03 165	. . . . . . . . 9 |- ((x e. B -> -. x e. A) <-> (x e. A -> -. x e. B))
10	8, 9	syl6bb 536	. . . . . . . 8 |- ((B (_ On /\ A e. B) -> ((x e. B -> A (_ x) <-> (x e. A -> -. x e. B)))
11	10	ralbidv2 1665	. . . . . . 7 |- ((B (_ On /\ A e. B) -> (A.x e. B A (_ x <-> A.x e. A -. x e. B))
12		ssint 2549	. . . . . . 7 |- (A (_ |^|B <-> A.x e. B A (_ x)
13	11, 12	syl5bb 532	. . . . . 6 |- ((B (_ On /\ A e. B) -> (A (_ |^|B <-> A.x e. A -. x e. B))
14	13	biimprd 154	. . . . 5 |- ((B (_ On /\ A e. B) -> (A.x e. A -. x e. B -> A (_ |^|B))
15	14	ex 373	. . . 4 |- (B (_ On -> (A e. B -> (A.x e. A -. x e. B -> A (_ |^|B)))
16	15	imp3a 361	. . 3 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A (_ |^|B))
17		intss1 2548	. . . . 5 |- (A e. B -> |^|B (_ A)
18	17	a1i 8	. . . 4 |- (B (_ On -> (A e. B -> |^|B (_ A))
19	18	adantrd 391	. . 3 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> |^|B (_ A))
20	16, 19	jcad 600	. 2 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> (A (_ |^|B /\ |^|B (_ A)))
21		eqss 2077	. 2 |- (A = |^|B <-> (A (_ |^|B /\ |^|B (_ A))
22	20, 21	syl6ibr 213	1 |- (B (_ On -> ((A e. B /\ A.x e. A -. x e. B) -> A = |^|B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047  |^|cint 2533  Oncon0 2948

This theorem is referenced by:  oneqmin 3018  alephval2 4902  alephval3 4903  cfsuc 4915

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952

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