Theorem wefrc 2938

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 2931	. . 3 |- (B (_ A -> (E We A -> E We B))
2		ineq2 2207	. . . . . . . . . . 11 |- (x = y -> (B i^i x) = (B i^i y))
3	2	eqeq1d 1480	. . . . . . . . . 10 |- (x = y -> ((B i^i x) = (/) <-> (B i^i y) = (/)))
4	3	rcla4ev 1873	. . . . . . . . 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 2226	. . . . . . . . . . 11 |- (B i^i y) (_ B
8		visset 1809	. . . . . . . . . . . . . . 15 |- y e. V
9	8	inex2 2712	. . . . . . . . . . . . . 14 |- (B i^i y) e. V
10	9	epfrc 2928	. . . . . . . . . . . . 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 2934	. . . . . . . . . . . . 13 |- (E We B -> E Fr B)
12	10, 11	syl3an1 858	. . . . . . . . . . . 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 831	. . . . . . . . . . 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 2203	. . . . . . . . . . . . 13 |- (x e. (B i^i y) <-> (x e. B /\ x e. y))
16	15	anbi1i 481	. . . . . . . . . . . 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 1669	. . . . . . . . . 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 2937	. . . . . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((y e. B /\ z e. B /\ x e. B) <-> ((y e. B /\ z e. B) /\ x e. B))
25		3anrot 779	. . . . . . . . . . . . . . . . . . . . . . 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 2203	. . . . . . . . . . . . . . . . 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 1715	. . . . . . . . . . . . 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 2055	. . . . . . . . . . . . 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 2050	. . . . . . . . . . . . . . . 16 |- ((B i^i x) (_ y <-> (B i^i x) = ((B i^i x) i^i y))
40		in23 2221	. . . . . . . . . . . . . . . . 17 |- ((B i^i x) i^i y) = ((B i^i y) i^i x)
41	40	eqeq2i 1482	. . . . . . . . . . . . . . . 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 1480	. . . . . . . . . . . . 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 1734	. . . . . . . 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 1632	. . . . . 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 1212	. . . 4 |- (E We B -> (E.y y e. B -> E.x e. B (B i^i x) = (/)))
54		ne0 2284	. . . 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 826	1 |- ((E We A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  E.wex 978   =/= wne 1582  A.wral 1642  E.wrex 1643   i^i cin 2042   (_ wss 2043  (/)c0 2276  Ecep 2825   Fr wfr 2910   We wwe 2911

This theorem is referenced by:  tz7.5 2964

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929

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