Theorem omsubsdomlem1OLD 15388

Description: Lemma for omsubsdom 5881. (Moved to omsubsdomlem1 5879 in main set.mm and may be deleted by mathbox owner, JGH. --NM 26-Aug-2011.)

Assertion

Ref	Expression
omsubsdomlem1OLD	|- (A e. On -> (aleph` A) ~< (aleph` suc A))

Proof of Theorem omsubsdomlem1OLD

Step	Hyp	Ref	Expression
1		alephon 5876	. . . . 5 |- (aleph` A) e. On
2		onsdom 5694	. . . . 5 |- ((aleph` A) e. On -> E.z e. On (aleph` A) ~< z)
3	1, 2	ax-mp 7	. . . 4 |- E.z e. On (aleph` A) ~< z
4	3	a1i 8	. . 3 |- (A e. On -> E.z e. On (aleph` A) ~< z)
5		ax-17 1317	. . . . 5 |- (m e. (aleph` A) -> A.z m e. (aleph` A))
6		ax-17 1317	. . . . 5 |- (m e. ~< -> A.z m e. ~< )
7		hbrab1 2257	. . . . . 6 |- (m e. {z e. On | (aleph` A) ~< z} -> A.z m e. {z e. On | (aleph` A) ~< z})
8	7	hbint 3225	. . . . 5 |- (m e. |^|{z e. On | (aleph` A) ~< z} -> A.z m e. |^|{z e. On | (aleph` A) ~< z})
9	5, 6, 8	hbbr 3381	. . . 4 |- ((aleph` A) ~< |^|{z e. On | (aleph` A) ~< z} -> A.z(aleph` A) ~< |^|{z e. On | (aleph` A) ~< z})
10		breq2 3342	. . . 4 |- (z = |^|{z e. On | (aleph` A) ~< z} -> ((aleph` A) ~< z <-> (aleph` A) ~< |^|{z e. On | (aleph` A) ~< z}))
11	9, 10	onminsb 3879	. . 3 |- (E.z e. On (aleph` A) ~< z -> (aleph` A) ~< |^|{z e. On | (aleph` A) ~< z})
12	4, 11	syl 12	. 2 |- (A e. On -> (aleph` A) ~< |^|{z e. On | (aleph` A) ~< z})
13		omsubsuc 5877	. 2 |- (A e. On -> (aleph` suc A) = |^|{z e. On | (aleph` A) ~< z})
14	12, 13	breqtrrd 3363	1 |- (A e. On -> (aleph` A) ~< (aleph` suc A))

Syntax hints:   -> wi 3   e. wcel 1300  E.wrex 2106  {crab 2108  |^|cint 3214   class class class wbr 3338  Oncon0 3657  suc csuc 3659  ` cfv 3998   ~< csdm 5425  alephcale 5860

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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