Theorem tz7.7 2969

Description: Proposition 7.7 of [TakeutiZaring] p. 37.

Assertion

Ref	Expression
tz7.7	|- ((Ord A /\ Tr B) -> (B e. A <-> (B (_ A /\ B =/= A)))

Proof of Theorem tz7.7

Step	Hyp	Ref	Expression
1		tz7.2 2927	. . . . 5 |- ((Tr A /\ E Fr A /\ B e. A) -> (B (_ A /\ B =/= A))
2	1	3exp 831	. . . 4 |- (Tr A -> (E Fr A -> (B e. A -> (B (_ A /\ B =/= A))))
3		ordtr 2958	. . . 4 |- (Ord A -> Tr A)
4		ordfr 2959	. . . 4 |- (Ord A -> E Fr A)
5	2, 3, 4	sylc 68	. . 3 |- (Ord A -> (B e. A -> (B (_ A /\ B =/= A)))
6	5	adantr 389	. 2 |- ((Ord A /\ Tr B) -> (B e. A -> (B (_ A /\ B =/= A)))
7		trss 2685	. . . . . . . . . . . . . . . . . . . 20 |- (Tr A -> (x e. A -> x (_ A))
8		eldifi 2159	. . . . . . . . . . . . . . . . . . . 20 |- (x e. (A \ B) -> x e. A)
9	7, 8	syl5 21	. . . . . . . . . . . . . . . . . . 19 |- (Tr A -> (x e. (A \ B) -> x (_ A))
10		difin0ss 2329	. . . . . . . . . . . . . . . . . . . 20 |- (((A \ B) i^i x) = (/) -> (x (_ A -> x (_ B))
11	10	com12 11	. . . . . . . . . . . . . . . . . . 19 |- (x (_ A -> (((A \ B) i^i x) = (/) -> x (_ B))
12	9, 11	syl6 22	. . . . . . . . . . . . . . . . . 18 |- (Tr A -> (x e. (A \ B) -> (((A \ B) i^i x) = (/) -> x (_ B)))
13	3, 12	syl 10	. . . . . . . . . . . . . . . . 17 |- (Ord A -> (x e. (A \ B) -> (((A \ B) i^i x) = (/) -> x (_ B)))
14	13	ad2antrr 404	. . . . . . . . . . . . . . . 16 |- (((Ord A /\ Tr B) /\ B (_ A) -> (x e. (A \ B) -> (((A \ B) i^i x) = (/) -> x (_ B)))
15	14	imp32 363	. . . . . . . . . . . . . . 15 |- ((((Ord A /\ Tr B) /\ B (_ A) /\ (x e. (A \ B) /\ ((A \ B) i^i x) = (/))) -> x (_ B)
16		eleq1 1532	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y = x -> (y e. B <-> x e. B))
17	16	biimpcd 155	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y e. B -> (y = x -> x e. B))
18		eldifn 2160	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x e. (A \ B) -> -. x e. B)
19	17, 18	nsyli 121	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y e. B -> (x e. (A \ B) -> -. y = x))
20	19	imp 350	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y e. B /\ x e. (A \ B)) -> -. y = x)
21	20	adantll 392	. . . . . . . . . . . . . . . . . . . . . 22 |- (((B (_ A /\ y e. B) /\ x e. (A \ B)) -> -. y = x)
22	21	adantl 388	. . . . . . . . . . . . . . . . . . . . 21 |- (((Ord A /\ Tr B) /\ ((B (_ A /\ y e. B) /\ x e. (A \ B))) -> -. y = x)
23		trel 2683	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (Tr B -> ((x e. y /\ y e. B) -> x e. B))
24	23	exp3a 375	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (Tr B -> (x e. y -> (y e. B -> x e. B)))
25	24	com23 32	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (Tr B -> (y e. B -> (x e. y -> x e. B)))
26	25	imp 350	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((Tr B /\ y e. B) -> (x e. y -> x e. B))
27	26, 18	nsyli 121	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((Tr B /\ y e. B) -> (x e. (A \ B) -> -. x e. y))
28	27	ex 373	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (Tr B -> (y e. B -> (x e. (A \ B) -> -. x e. y)))
29	28	adantld 390	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Tr B -> ((B (_ A /\ y e. B) -> (x e. (A \ B) -> -. x e. y)))
30	29	imp32 363	. . . . . . . . . . . . . . . . . . . . . 22 |- ((Tr B /\ ((B (_ A /\ y e. B) /\ x e. (A \ B))) -> -. x e. y)
31	30	adantll 392	. . . . . . . . . . . . . . . . . . . . 21 |- (((Ord A /\ Tr B) /\ ((B (_ A /\ y e. B) /\ x e. (A \ B))) -> -. x e. y)
32		wecmpep 2937	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((E We A /\ (y e. A /\ x e. A)) -> (y e. x \/ y = x \/ x e. y))
33		ordwe 2957	. . . . . . . . . . . . . . . . . . . . . . 23 |- (Ord A -> E We A)
34		ssel2 2061	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((B (_ A /\ y e. B) -> y e. A)
35	34, 8	anim12i 333	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((B (_ A /\ y e. B) /\ x e. (A \ B)) -> (y e. A /\ x e. A))
36	32, 33, 35	syl2an 454	. . . . . . . . . . . . . . . . . . . . . 22 |- ((Ord A /\ ((B (_ A /\ y e. B) /\ x e. (A \ B))) -> (y e. x \/ y = x \/ x e. y))
37	36	adantlr 393	. . . . . . . . . . . . . . . . . . . . 21 |- (((Ord A /\ Tr B) /\ ((B (_ A /\ y e. B) /\ x e. (A \ B))) -> (y e. x \/ y = x \/ x e. y))
38	22, 31, 37	ecase23d 921	. . . . . . . . . . . . . . . . . . . 20 |- (((Ord A /\ Tr B) /\ ((B (_ A /\ y e. B) /\ x e. (A \ B))) -> y e. x)
39	38	exp44 385	. . . . . . . . . . . . . . . . . . 19 |- ((Ord A /\ Tr B) -> (B (_ A -> (y e. B -> (x e. (A \ B) -> y e. x))))
40	39	com34 36	. . . . . . . . . . . . . . . . . 18 |- ((Ord A /\ Tr B) -> (B (_ A -> (x e. (A \ B) -> (y e. B -> y e. x))))
41	40	imp31 362	. . . . . . . . . . . . . . . . 17 |- ((((Ord A /\ Tr B) /\ B (_ A) /\ x e. (A \ B)) -> (y e. B -> y e. x))
42	41	ssrdv 2067	. . . . . . . . . . . . . . . 16 |- ((((Ord A /\ Tr B) /\ B (_ A) /\ x e. (A \ B)) -> B (_ x)
43	42	adantrr 395	. . . . . . . . . . . . . . 15 |- ((((Ord A /\ Tr B) /\ B (_ A) /\ (x e. (A \ B) /\ ((A \ B) i^i x) = (/))) -> B (_ x)
44	15, 43	eqssd 2076	. . . . . . . . . . . . . 14 |- ((((Ord A /\ Tr B) /\ B (_ A) /\ (x e. (A \ B) /\ ((A \ B) i^i x) = (/))) -> x = B)
45	8	ad2antrl 406	. . . . . . . . . . . . . 14 |- ((((Ord A /\ Tr B) /\ B (_ A) /\ (x e. (A \ B) /\ ((A \ B) i^i x) = (/))) -> x e. A)
46	44, 45	eqeltrrd 1547	. . . . . . . . . . . . 13 |- ((((Ord A /\ Tr B) /\ B (_ A) /\ (x e. (A \ B) /\ ((A \ B) i^i x) = (/))) -> B e. A)
47	46	exp32 377	. . . . . . . . . . . 12 |- (((Ord A /\ Tr B) /\ B (_ A) -> (x e. (A \ B) -> (((A \ B) i^i x) = (/) -> B e. A)))
48	47	r19.23adv 1744	. . . . . . . . . . 11 |- (((Ord A /\ Tr B) /\ B (_ A) -> (E.x e. (A \ B)((A \ B) i^i x) = (/) -> B e. A))
49		difss 2164	. . . . . . . . . . . 12 |- (A \ B) (_ A
50		tz7.5 2965	. . . . . . . . . . . 12 |- ((Ord A /\ (A \ B) (_ A /\ (A \ B) =/= (/)) -> E.x e. (A \ B)((A \ B) i^i x) = (/))
51	49, 50	mp3an2 903	. . . . . . . . . . 11 |- ((Ord A /\ (A \ B) =/= (/)) -> E.x e. (A \ B)((A \ B) i^i x) = (/))
52	48, 51	syl5 21	. . . . . . . . . 10 |- (((Ord A /\ Tr B) /\ B (_ A) -> ((Ord A /\ (A \ B) =/= (/)) -> B e. A))
53	52	exp4b 379	. . . . . . . . 9 |- ((Ord A /\ Tr B) -> (B (_ A -> (Ord A -> ((A \ B) =/= (/) -> B e. A))))
54	53	com23 32	. . . . . . . 8 |- ((Ord A /\ Tr B) -> (Ord A -> (B (_ A -> ((A \ B) =/= (/) -> B e. A))))
55	54	adantrd 391	. . . . . . 7 |- ((Ord A /\ Tr B) -> ((Ord A /\ Tr B) -> (B (_ A -> ((A \ B) =/= (/) -> B e. A))))
56	55	pm2.43i 64	. . . . . 6 |- ((Ord A /\ Tr B) -> (B (_ A -> ((A \ B) =/= (/) -> B e. A)))
57		pssdifn0 2326	. . . . . 6 |- ((B (_ A /\ B =/= A) -> (A \ B) =/= (/))
58	56, 57	syl7 23	. . . . 5 |- ((Ord A /\ Tr B) -> (B (_ A -> ((B (_ A /\ B =/= A) -> B e. A)))
59	58	exp4a 378	. . . 4 |- ((Ord A /\ Tr B) -> (B (_ A -> (B (_ A -> (B =/= A -> B e. A))))
60	59	pm2.43d 65	. . 3 |- ((Ord A /\ Tr B) -> (B (_ A -> (B =/= A -> B e. A)))
61	60	imp3a 361	. 2 |- ((Ord A /\ Tr B) -> ((B (_ A /\ B =/= A) -> B e. A))
62	6, 61	impbid 515	1 |- ((Ord A /\ Tr B) -> (B e. A <-> (B (_ A /\ B =/= A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   \/ w3o 773   = wceq 955   e. wcel 957   =/= wne 1583  E.wrex 1644   \ cdif 2041   i^i cin 2043   (_ wss 2044  (/)c0 2277  Tr wtr 2676  Ecep 2826   Fr wfr 2911   We wwe 2912  Ord word 2943

This theorem is referenced by:  ordelssne 2970

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947

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