Theorem onsdomOLD 15385

Description: Any ordinal number is strictly dominated by some other ordinal number. (Moved to onsdom 5694 in main set.mm and may be deleted by mathbox owner, JGH. --NM 26-Aug-2011.)

Assertion

Ref	Expression
onsdomOLD	|- (A e. On -> E.x e. On A ~< x)

Distinct variable group:   x,A

Proof of Theorem onsdomOLD

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- {y e. On | y ~<_ A} = {y e. On | y ~<_ A}
2	1	hartog 5693	. 2 |- (A e. On -> {y e. On | y ~<_ A} e. On)
3		brsdom 5440	. . 3 |- (A ~< {y e. On | y ~<_ A} <-> (A ~<_ {y e. On | y ~<_ A} /\ -. A ~~ {y e. On | y ~<_ A}))
4		onelon 3683	. . . . . . . 8 |- ((A e. On /\ z e. A) -> z e. On)
5	4	ex 402	. . . . . . 7 |- (A e. On -> (z e. A -> z e. On))
6		onelss 3705	. . . . . . . 8 |- (A e. On -> (z e. A -> z C_ A))
7		visset 2295	. . . . . . . . 9 |- z e. _V
8		ssdomg 5467	. . . . . . . . 9 |- (z e. _V -> (z C_ A -> z ~<_ A))
9	7, 8	ax-mp 7	. . . . . . . 8 |- (z C_ A -> z ~<_ A)
10	6, 9	syl6 25	. . . . . . 7 |- (A e. On -> (z e. A -> z ~<_ A))
11	5, 10	jcad 661	. . . . . 6 |- (A e. On -> (z e. A -> (z e. On /\ z ~<_ A)))
12		breq1 3341	. . . . . . 7 |- (y = z -> (y ~<_ A <-> z ~<_ A))
13	12	elrab 2414	. . . . . 6 |- (z e. {y e. On | y ~<_ A} <-> (z e. On /\ z ~<_ A))
14	11, 13	syl6ibr 230	. . . . 5 |- (A e. On -> (z e. A -> z e. {y e. On | y ~<_ A}))
15	14	ssrdv 2622	. . . 4 |- (A e. On -> A C_ {y e. On | y ~<_ A})
16		ssdomg 5467	. . . 4 |- (A e. On -> (A C_ {y e. On | y ~<_ A} -> A ~<_ {y e. On | y ~<_ A}))
17	15, 16	mpd 29	. . 3 |- (A e. On -> A ~<_ {y e. On | y ~<_ A})
18		eloni 3667	. . . . . 6 |- ({y e. On | y ~<_ A} e. On -> Ord {y e. On | y ~<_ A})
19		ordirr 3676	. . . . . 6 |- (Ord {y e. On | y ~<_ A} -> -. {y e. On | y ~<_ A} e. {y e. On | y ~<_ A})
20	2, 18, 19	3syl 24	. . . . 5 |- (A e. On -> -. {y e. On | y ~<_ A} e. {y e. On | y ~<_ A})
21	2	a1d 15	. . . . . . 7 |- (A e. On -> ({y e. On | y ~<_ A} ~<_ A -> {y e. On | y ~<_ A} e. On))
22	21	ancrd 323	. . . . . 6 |- (A e. On -> ({y e. On | y ~<_ A} ~<_ A -> ({y e. On | y ~<_ A} e. On /\ {y e. On | y ~<_ A} ~<_ A)))
23		breq1 3341	. . . . . . . 8 |- (z = {y e. On | y ~<_ A} -> (z ~<_ A <-> {y e. On | y ~<_ A} ~<_ A))
24	23	elrab 2414	. . . . . . 7 |- ({y e. On | y ~<_ A} e. {z e. On | z ~<_ A} <-> ({y e. On | y ~<_ A} e. On /\ {y e. On | y ~<_ A} ~<_ A))
25		breq1 3341	. . . . . . . . 9 |- (z = y -> (z ~<_ A <-> y ~<_ A))
26	25	cbvrabv 2422	. . . . . . . 8 |- {z e. On | z ~<_ A} = {y e. On | y ~<_ A}
27	26	eleq2i 1961	. . . . . . 7 |- ({y e. On | y ~<_ A} e. {z e. On | z ~<_ A} <-> {y e. On | y ~<_ A} e. {y e. On | y ~<_ A})
28	24, 27	bitr3i 192	. . . . . 6 |- (({y e. On | y ~<_ A} e. On /\ {y e. On | y ~<_ A} ~<_ A) <-> {y e. On | y ~<_ A} e. {y e. On | y ~<_ A})
29	22, 28	syl6ib 229	. . . . 5 |- (A e. On -> ({y e. On | y ~<_ A} ~<_ A -> {y e. On | y ~<_ A} e. {y e. On | y ~<_ A}))
30	20, 29	mtod 123	. . . 4 |- (A e. On -> -. {y e. On | y ~<_ A} ~<_ A)
31		ensymg 5470	. . . . . 6 |- ({y e. On | y ~<_ A} e. On -> (A ~~ {y e. On | y ~<_ A} -> {y e. On | y ~<_ A} ~~ A))
32	2, 31	syl 12	. . . . 5 |- (A e. On -> (A ~~ {y e. On | y ~<_ A} -> {y e. On | y ~<_ A} ~~ A))
33		endom 5444	. . . . 5 |- ({y e. On | y ~<_ A} ~~ A -> {y e. On | y ~<_ A} ~<_ A)
34	32, 33	syl6 25	. . . 4 |- (A e. On -> (A ~~ {y e. On | y ~<_ A} -> {y e. On | y ~<_ A} ~<_ A))
35	30, 34	mtod 123	. . 3 |- (A e. On -> -. A ~~ {y e. On | y ~<_ A})
36	3, 17, 35	sylanbrc 527	. 2 |- (A e. On -> A ~< {y e. On | y ~<_ A})
37		breq2 3342	. . 3 |- (x = {y e. On | y ~<_ A} -> (A ~< x <-> A ~< {y e. On | y ~<_ A}))
38	37	rcla4ev 2381	. 2 |- (({y e. On | y ~<_ A} e. On /\ A ~< {y e. On | y ~<_ A}) -> E.x e. On A ~< x)
39	2, 36, 38	syl11anc 524	1 |- (A e. On -> E.x e. On A ~< x)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   e. wcel 1300  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  Ord word 3656  Oncon0 3657   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425

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

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-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-suc 3663  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-er 5318  df-en 5427  df-dom 5428  df-sdom 5429

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