Theorem omsubsdomlem2 5880

Description: Lemma for omsubsdom 5881.

Assertion

Ref	Expression
omsubsdomlem2	|- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) ~< (aleph` B)))

Proof of Theorem omsubsdomlem2

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . 5 |- (k = suc A -> (aleph` k) = (aleph` suc A))
2	1	breq2d 3350	. . . 4 |- (k = suc A -> ((aleph` A) ~< (aleph` k) <-> (aleph` A) ~< (aleph` suc A)))
3		fveq2 4681	. . . . 5 |- (k = j -> (aleph` k) = (aleph` j))
4	3	breq2d 3350	. . . 4 |- (k = j -> ((aleph` A) ~< (aleph` k) <-> (aleph` A) ~< (aleph` j)))
5		fveq2 4681	. . . . 5 |- (k = suc j -> (aleph` k) = (aleph` suc j))
6	5	breq2d 3350	. . . 4 |- (k = suc j -> ((aleph` A) ~< (aleph` k) <-> (aleph` A) ~< (aleph` suc j)))
7		fveq2 4681	. . . . 5 |- (k = B -> (aleph` k) = (aleph` B))
8	7	breq2d 3350	. . . 4 |- (k = B -> ((aleph` A) ~< (aleph` k) <-> (aleph` A) ~< (aleph` B)))
9		omsubsdomlem1 5879	. . . 4 |- (A e. On -> (aleph` A) ~< (aleph` suc A))
10		sdomtr 5537	. . . . . 6 |- (((aleph` A) ~< (aleph` j) /\ (aleph` j) ~< (aleph` suc j)) -> (aleph` A) ~< (aleph` suc j))
11		omsubsdomlem1 5879	. . . . . . 7 |- (j e. On -> (aleph` j) ~< (aleph` suc j))
12	11	adantr 425	. . . . . 6 |- ((j e. On /\ A e. j) -> (aleph` j) ~< (aleph` suc j))
13	10, 12	sylan2 500	. . . . 5 |- (((aleph` A) ~< (aleph` j) /\ (j e. On /\ A e. j)) -> (aleph` A) ~< (aleph` suc j))
14	13	expcom 403	. . . 4 |- ((j e. On /\ A e. j) -> ((aleph` A) ~< (aleph` j) -> (aleph` A) ~< (aleph` suc j)))
15		hbra1 2147	. . . . . . . . . 10 |- (A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> A.jA.j e. k (A e. j -> (aleph` A) ~< (aleph` j)))
16		ax-17 1317	. . . . . . . . . . 11 |- (m e. (aleph` A) -> A.j m e. (aleph` A))
17		ax-17 1317	. . . . . . . . . . 11 |- (m e. ~< -> A.j m e. ~< )
18		hbiu1 3281	. . . . . . . . . . 11 |- (m e. U_j e. k (aleph` j) -> A.j m e. U_j e. k (aleph` j))
19	16, 17, 18	hbbr 3381	. . . . . . . . . 10 |- ((aleph` A) ~< U_j e. k (aleph` j) -> A.j(aleph` A) ~< U_j e. k (aleph` j))
20	15, 19	hbim 1354	. . . . . . . . 9 |- ((A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> (aleph` A) ~< U_j e. k (aleph` j)) -> A.j(A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> (aleph` A) ~< U_j e. k (aleph` j)))
21		ra4 2155	. . . . . . . . . . . 12 |- (A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> (j e. k -> (A e. j -> (aleph` A) ~< (aleph` j))))
22	21	com12 14	. . . . . . . . . . 11 |- (j e. k -> (A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> (A e. j -> (aleph` A) ~< (aleph` j))))
23	22	adantl 424	. . . . . . . . . 10 |- ((A e. j /\ j e. k) -> (A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> (A e. j -> (aleph` A) ~< (aleph` j))))
24		visset 2295	. . . . . . . . . . . . . . . . 17 |- k e. _V
25		fvex 4689	. . . . . . . . . . . . . . . . 17 |- (aleph` j) e. _V
26	24, 25	iunex 4839	. . . . . . . . . . . . . . . 16 |- U_j e. k (aleph` j) e. _V
27		sdomdomtr 5532	. . . . . . . . . . . . . . . 16 |- (U_j e. k (aleph` j) e. _V -> (((aleph` A) ~< (aleph` j) /\ (aleph` j) ~<_ U_j e. k (aleph` j)) -> (aleph` A) ~< U_j e. k (aleph` j)))
28	26, 27	ax-mp 7	. . . . . . . . . . . . . . 15 |- (((aleph` A) ~< (aleph` j) /\ (aleph` j) ~<_ U_j e. k (aleph` j)) -> (aleph` A) ~< U_j e. k (aleph` j))
29	25	a1i 8	. . . . . . . . . . . . . . . 16 |- (j e. k -> (aleph` j) e. _V)
30		ssiun2 3295	. . . . . . . . . . . . . . . 16 |- (j e. k -> (aleph` j) C_ U_j e. k (aleph` j))
31		ssdomg 5467	. . . . . . . . . . . . . . . 16 |- ((aleph` j) e. _V -> ((aleph` j) C_ U_j e. k (aleph` j) -> (aleph` j) ~<_ U_j e. k (aleph` j)))
32	29, 30, 31	sylc 83	. . . . . . . . . . . . . . 15 |- (j e. k -> (aleph` j) ~<_ U_j e. k (aleph` j))
33	28, 32	sylan2 500	. . . . . . . . . . . . . 14 |- (((aleph` A) ~< (aleph` j) /\ j e. k) -> (aleph` A) ~< U_j e. k (aleph` j))
34	33	expcom 403	. . . . . . . . . . . . 13 |- (j e. k -> ((aleph` A) ~< (aleph` j) -> (aleph` A) ~< U_j e. k (aleph` j)))
35	34	imim2d 28	. . . . . . . . . . . 12 |- (j e. k -> ((A e. j -> (aleph` A) ~< (aleph` j)) -> (A e. j -> (aleph` A) ~< U_j e. k (aleph` j))))
36	35	com23 36	. . . . . . . . . . 11 |- (j e. k -> (A e. j -> ((A e. j -> (aleph` A) ~< (aleph` j)) -> (aleph` A) ~< U_j e. k (aleph` j))))
37	36	impcom 378	. . . . . . . . . 10 |- ((A e. j /\ j e. k) -> ((A e. j -> (aleph` A) ~< (aleph` j)) -> (aleph` A) ~< U_j e. k (aleph` j)))
38	23, 37	syld 30	. . . . . . . . 9 |- ((A e. j /\ j e. k) -> (A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> (aleph` A) ~< U_j e. k (aleph` j)))
39	20, 38	19.23ai 1412	. . . . . . . 8 |- (E.j(A e. j /\ j e. k) -> (A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> (aleph` A) ~< U_j e. k (aleph` j)))
40	39	imp 377	. . . . . . 7 |- ((E.j(A e. j /\ j e. k) /\ A.j e. k (A e. j -> (aleph` A) ~< (aleph` j))) -> (aleph` A) ~< U_j e. k (aleph` j))
41		limuni 3724	. . . . . . . . . 10 |- (Lim k -> k = U.k)
42	41	eleq2d 1964	. . . . . . . . 9 |- (Lim k -> (A e. k <-> A e. U.k))
43	42	biimpa 460	. . . . . . . 8 |- ((Lim k /\ A e. k) -> A e. U.k)
44		eluni 3180	. . . . . . . 8 |- (A e. U.k <-> E.j(A e. j /\ j e. k))
45	43, 44	sylib 215	. . . . . . 7 |- ((Lim k /\ A e. k) -> E.j(A e. j /\ j e. k))
46	40, 45	sylan 497	. . . . . 6 |- (((Lim k /\ A e. k) /\ A.j e. k (A e. j -> (aleph` A) ~< (aleph` j))) -> (aleph` A) ~< U_j e. k (aleph` j))
47		alephlim 5875	. . . . . . . . 9 |- ((k e. _V /\ Lim k) -> (aleph` k) = U_j e. k (aleph` j))
48	24, 47	mpan 759	. . . . . . . 8 |- (Lim k -> (aleph` k) = U_j e. k (aleph` j))
49	48	ad2antrr 440	. . . . . . 7 |- (((Lim k /\ A e. k) /\ A.j e. k (A e. j -> (aleph` A) ~< (aleph` j))) -> (aleph` k) = U_j e. k (aleph` j))
50	49	eqcomd 1889	. . . . . 6 |- (((Lim k /\ A e. k) /\ A.j e. k (A e. j -> (aleph` A) ~< (aleph` j))) -> U_j e. k (aleph` j) = (aleph` k))
51	46, 50	breqtrd 3361	. . . . 5 |- (((Lim k /\ A e. k) /\ A.j e. k (A e. j -> (aleph` A) ~< (aleph` j))) -> (aleph` A) ~< (aleph` k))
52	51	ex 402	. . . 4 |- ((Lim k /\ A e. k) -> (A.j e. k (A e. j -> (aleph` A) ~< (aleph` j)) -> (aleph` A) ~< (aleph` k)))
53	2, 4, 6, 8, 9, 14, 52	tfindsg2 3945	. . 3 |- ((B e. On /\ A e. B) -> (aleph` A) ~< (aleph` B))
54	53	ex 402	. 2 |- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))
55	54	adantl 424	1 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) ~< (aleph` B)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  _Vcvv 2292   C_ wss 2593  U.cuni 3177  U_ciun 3255   class class class wbr 3338  Oncon0 3657  Lim wlim 3658  suc csuc 3659  ` cfv 3998   ~<_ cdom 5424   ~< csdm 5425  alephcale 5860

This theorem is referenced by:  omsubsdom 5881  omsubsdomOLD 15390

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-iso 4015  df-oprab 4887  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-aleph 5863

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