Theorem wefrc 2933

Description: A non-empty (possibly proper) subclass of a class well-ordered by E has a minimal element. Special case of Proposition 6.26 of [TakeutiZaring] p. 31.

Assertion

Ref	Expression
wefrc	|- ((E We A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Distinct variable group:   x,B

Proof of Theorem wefrc

Step	Hyp	Ref	Expression
1		wess 2926	. . 3 |- (B (_ A -> (E We A -> E We B))
2		ineq2 2201	. . . . . . . . . . 11 |- (x = y -> (B i^i x) = (B i^i y))
3	2	eqeq1d 1475	. . . . . . . . . 10 |- (x = y -> ((B i^i x) = (/) <-> (B i^i y) = (/)))
4	3	rcla4ev 1868	. . . . . . . . 9 |- ((y e. B /\ (B i^i y) = (/)) -> E.x e. B (B i^i x) = (/))
5	4	ex 373	. . . . . . . 8 |- (y e. B -> ((B i^i y) = (/) -> E.x e. B (B i^i x) = (/)))
6	5	adantl 388	. . . . . . 7 |- ((E We B /\ y e. B) -> ((B i^i y) = (/) -> E.x e. B (B i^i x) = (/)))
7		inss1 2220	. . . . . . . . . . 11 |- (B i^i y) (_ B
8		visset 1804	. . . . . . . . . . . . . . 15 |- y e. V
9	8	inex2 2707	. . . . . . . . . . . . . 14 |- (B i^i y) e. V
10	9	epfrc 2923	. . . . . . . . . . . . 13 |- ((E Fr B /\ (B i^i y) (_ B /\ (B i^i y) =/= (/)) -> E.x e. (B i^i y)((B i^i y) i^i x) = (/))
11		wefr 2929	. . . . . . . . . . . . 13 |- (E We B -> E Fr B)
12	10, 11	syl3an1 857	. . . . . . . . . . . 12 |- ((E We B /\ (B i^i y) (_ B /\ (B i^i y) =/= (/)) -> E.x e. (B i^i y)((B i^i y) i^i x) = (/))
13	12	3exp 830	. . . . . . . . . . 11 |- (E We B -> ((B i^i y) (_ B -> ((B i^i y) =/= (/) -> E.x e. (B i^i y)((B i^i y) i^i x) = (/))))
14	7, 13	mpi 44	. . . . . . . . . 10 |- (E We B -> ((B i^i y) =/= (/) -> E.x e. (B i^i y)((B i^i y) i^i x) = (/)))
15		elin 2197	. . . . . . . . . . . . 13 |- (x e. (B i^i y) <-> (x e. B /\ x e. y))
16	15	anbi1i 480	. . . . . . . . . . . 12 |- ((x e. (B i^i y) /\ ((B i^i y) i^i x) = (/)) <-> ((x e. B /\ x e. y) /\ ((B i^i y) i^i x) = (/)))
17		anass 439	. . . . . . . . . . . 12 |- (((x e. B /\ x e. y) /\ ((B i^i y) i^i x) = (/)) <-> (x e. B /\ (x e. y /\ ((B i^i y) i^i x) = (/))))
18	16, 17	bitr 173	. . . . . . . . . . 11 |- ((x e. (B i^i y) /\ ((B i^i y) i^i x) = (/)) <-> (x e. B /\ (x e. y /\ ((B i^i y) i^i x) = (/))))
19	18	rexbii2 1664	. . . . . . . . . 10 |- (E.x e. (B i^i y)((B i^i y) i^i x) = (/) <-> E.x e. B (x e. y /\ ((B i^i y) i^i x) = (/)))
20	14, 19	syl6ib 212	. . . . . . . . 9 |- (E We B -> ((B i^i y) =/= (/) -> E.x e. B (x e. y /\ ((B i^i y) i^i x) = (/))))
21	20	adantr 389	. . . . . . . 8 |- ((E We B /\ y e. B) -> ((B i^i y) =/= (/) -> E.x e. B (x e. y /\ ((B i^i y) i^i x) = (/))))
22		wetrep 2932	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((E We B /\ (z e. B /\ x e. B /\ y e. B)) -> ((z e. x /\ x e. y) -> z e. y))
23	22	exp3a 375	. . . . . . . . . . . . . . . . . . . . . 22 |- ((E We B /\ (z e. B /\ x e. B /\ y e. B)) -> (z e. x -> (x e. y -> z e. y)))
24		df-3an 775	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y e. B /\ z e. B /\ x e. B) <-> ((y e. B /\ z e. B) /\ x e. B))
25		3anrot 778	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y e. B /\ z e. B /\ x e. B) <-> (z e. B /\ x e. B /\ y e. B))
26	24, 25	bitr3 175	. . . . . . . . . . . . . . . . . . . . . 22 |- (((y e. B /\ z e. B) /\ x e. B) <-> (z e. B /\ x e. B /\ y e. B))
27	23, 26	sylan2b 452	. . . . . . . . . . . . . . . . . . . . 21 |- ((E We B /\ ((y e. B /\ z e. B) /\ x e. B)) -> (z e. x -> (x e. y -> z e. y)))
28	27	exp44 385	. . . . . . . . . . . . . . . . . . . 20 |- (E We B -> (y e. B -> (z e. B -> (x e. B -> (z e. x -> (x e. y -> z e. y))))))
29	28	imp 350	. . . . . . . . . . . . . . . . . . 19 |- ((E We B /\ y e. B) -> (z e. B -> (x e. B -> (z e. x -> (x e. y -> z e. y)))))
30	29	com34 36	. . . . . . . . . . . . . . . . . 18 |- ((E We B /\ y e. B) -> (z e. B -> (z e. x -> (x e. B -> (x e. y -> z e. y)))))
31	30	imp3a 361	. . . . . . . . . . . . . . . . 17 |- ((E We B /\ y e. B) -> ((z e. B /\ z e. x) -> (x e. B -> (x e. y -> z e. y))))
32		elin 2197	. . . . . . . . . . . . . . . . 17 |- (z e. (B i^i x) <-> (z e. B /\ z e. x))
33	31, 32	syl5ib 206	. . . . . . . . . . . . . . . 16 |- ((E We B /\ y e. B) -> (z e. (B i^i x) -> (x e. B -> (x e. y -> z e. y))))
34	33	imp4a 364	. . . . . . . . . . . . . . 15 |- ((E We B /\ y e. B) -> (z e. (B i^i x) -> ((x e. B /\ x e. y) -> z e. y)))
35	34	com23 32	. . . . . . . . . . . . . 14 |- ((E We B /\ y e. B) -> ((x e. B /\ x e. y) -> (z e. (B i^i x) -> z e. y)))
36	35	r19.21adv 1710	. . . . . . . . . . . . 13 |- ((E We B /\ y e. B) -> ((x e. B /\ x e. y) -> A.z e. (B i^i x)z e. y))
37		dfss3 2049	. . . . . . . . . . . . 13 |- ((B i^i x) (_ y <-> A.z e. (B i^i x)z e. y)
38	36, 37	syl6ibr 213	. . . . . . . . . . . 12 |- ((E We B /\ y e. B) -> ((x e. B /\ x e. y) -> (B i^i x) (_ y))
39		dfss 2044	. . . . . . . . . . . . . . . 16 |- ((B i^i x) (_ y <-> (B i^i x) = ((B i^i x) i^i y))
40		in23 2215	. . . . . . . . . . . . . . . . 17 |- ((B i^i x) i^i y) = ((B i^i y) i^i x)
41	40	eqeq2i 1477	. . . . . . . . . . . . . . . 16 |- ((B i^i x) = ((B i^i x) i^i y) <-> (B i^i x) = ((B i^i y) i^i x))
42	39, 41	bitr 173	. . . . . . . . . . . . . . 15 |- ((B i^i x) (_ y <-> (B i^i x) = ((B i^i y) i^i x))
43	42	biimp 151	. . . . . . . . . . . . . 14 |- ((B i^i x) (_ y -> (B i^i x) = ((B i^i y) i^i x))
44	43	eqeq1d 1475	. . . . . . . . . . . . 13 |- ((B i^i x) (_ y -> ((B i^i x) = (/) <-> ((B i^i y) i^i x) = (/)))
45	44	biimprd 154	. . . . . . . . . . . 12 |- ((B i^i x) (_ y -> (((B i^i y) i^i x) = (/) -> (B i^i x) = (/)))
46	38, 45	syl6 22	. . . . . . . . . . 11 |- ((E We B /\ y e. B) -> ((x e. B /\ x e. y) -> (((B i^i y) i^i x) = (/) -> (B i^i x) = (/))))
47	46	exp3a 375	. . . . . . . . . 10 |- ((E We B /\ y e. B) -> (x e. B -> (x e. y -> (((B i^i y) i^i x) = (/) -> (B i^i x) = (/)))))
48	47	imp4a 364	. . . . . . . . 9 |- ((E We B /\ y e. B) -> (x e. B -> ((x e. y /\ ((B i^i y) i^i x) = (/)) -> (B i^i x) = (/))))
49	48	r19.22dv 1729	. . . . . . . 8 |- ((E We B /\ y e. B) -> (E.x e. B (x e. y /\ ((B i^i y) i^i x) = (/)) -> E.x e. B (B i^i x) = (/)))
50	21, 49	syld 27	. . . . . . 7 |- ((E We B /\ y e. B) -> ((B i^i y) =/= (/) -> E.x e. B (B i^i x) = (/)))
51	6, 50	pm2.61dne 1627	. . . . . 6 |- ((E We B /\ y e. B) -> E.x e. B (B i^i x) = (/))
52	51	ex 373	. . . . 5 |- (E We B -> (y e. B -> E.x e. B (B i^i x) = (/)))
53	52	19.23adv 1209	. . . 4 |- (E We B -> (E.y y e. B -> E.x e. B (B i^i x) = (/)))
54		ne0 2278	. . . 4 |- (B =/= (/) <-> E.y y e. B)
55	53, 54	syl5ib 206	. . 3 |- (E We B -> (B =/= (/) -> E.x e. B (B i^i x) = (/)))
56	1, 55	syl6com 53	. 2 |- (E We A -> (B (_ A -> (B =/= (/) -> E.x e. B (B i^i x) = (/))))
57	56	3imp 825	1 |- ((E We A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  A.wral 1637  E.wrex 1638   i^i cin 2036   (_ wss 2037  (/)c0 2270  Ecep 2819   Fr wfr 2905   We wwe 2906

This theorem is referenced by:  tz7.5 2959

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-op 2406  df-br 2610  df-opab 2657  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924

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