Theorem axfelem1 14031

Description: Lemma for axfe (future) . Show a particular abstraction is an ordinal.

Hypotheses

Ref	Expression
axfelem1.1	|- X e. {1o, 2o}
axfelem1.2	|- C = |^|{a e. On | A.n e. F E.b e. a (n` b) =/= X}

Assertion

Ref	Expression
axfelem1	|- ((F C_ No /\ F e. A) -> C e. On)

Distinct variable groups:   F,a,b,n   X,a,b

Proof of Theorem axfelem1

Step	Hyp	Ref	Expression
1		rexeq 2267	. . . . . . 7 |- (a = suc U.( bday "F) -> (E.b e. a (n` b) =/= X <-> E.b e. suc U.( bday "F)(n` b) =/= X))
2	1	ralbidv 2123	. . . . . 6 |- (a = suc U.( bday "F) -> (A.n e. F E.b e. a (n` b) =/= X <-> A.n e. F E.b e. suc U.( bday "F)(n` b) =/= X))
3	2	rcla4ev 2381	. . . . 5 |- ((suc U.( bday "F) e. On /\ A.n e. F E.b e. suc U.( bday "F)(n` b) =/= X) -> E.a e. On A.n e. F E.b e. a (n` b) =/= X)
4		axfelem0 14030	. . . . 5 |- (F e. A -> suc U.( bday "F) e. On)
5		ssel2 2616	. . . . . . . . . . 11 |- ((F C_ No /\ n e. F) -> n e. No )
6		bdaydm 14015	. . . . . . . . . . 11 |- dom bday = No
7	5, 6	syl6eleqr 1982	. . . . . . . . . 10 |- ((F C_ No /\ n e. F) -> n e. dom bday )
8		simpr 350	. . . . . . . . . 10 |- ((F C_ No /\ n e. F) -> n e. F)
9		bdayfun 14013	. . . . . . . . . . 11 |- Fun bday
10		funfvima 4828	. . . . . . . . . . 11 |- ((Fun bday /\ n e. dom bday ) -> (n e. F -> ( bday ` n) e. ( bday "F)))
11	9, 10	mpan 759	. . . . . . . . . 10 |- (n e. dom bday -> (n e. F -> ( bday ` n) e. ( bday "F)))
12	7, 8, 11	sylc 83	. . . . . . . . 9 |- ((F C_ No /\ n e. F) -> ( bday ` n) e. ( bday "F))
13		elssuni 3206	. . . . . . . . 9 |- (( bday ` n) e. ( bday "F) -> ( bday ` n) C_ U.( bday "F))
14	12, 13	syl 12	. . . . . . . 8 |- ((F C_ No /\ n e. F) -> ( bday ` n) C_ U.( bday "F))
15		bdayelon 14017	. . . . . . . . 9 |- ( bday ` n) e. On
16		imassrn 4278	. . . . . . . . . . 11 |- ( bday "F) C_ ran bday
17		bdayrn 14014	. . . . . . . . . . 11 |- ran bday = On
18	16, 17	sseqtri 2649	. . . . . . . . . 10 |- ( bday "F) C_ On
19		ssorduni 3870	. . . . . . . . . 10 |- (( bday "F) C_ On -> Ord U.( bday "F))
20	18, 19	ax-mp 7	. . . . . . . . 9 |- Ord U.( bday "F)
21		ordsssuc 3756	. . . . . . . . 9 |- ((( bday ` n) e. On /\ Ord U.( bday "F)) -> (( bday ` n) C_ U.( bday "F) <-> ( bday ` n) e. suc U.( bday "F)))
22	15, 20, 21	mp2an 761	. . . . . . . 8 |- (( bday ` n) C_ U.( bday "F) <-> ( bday ` n) e. suc U.( bday "F))
23	14, 22	sylib 215	. . . . . . 7 |- ((F C_ No /\ n e. F) -> ( bday ` n) e. suc U.( bday "F))
24		axfelem1.1	. . . . . . . . 9 |- X e. {1o, 2o}
25	24	nosgnn0i 14000	. . . . . . . 8 |- (/) =/= X
26		axdenselem2 14020	. . . . . . . . . 10 |- (n e. No -> (n` ( bday ` n)) = (/))
27	5, 26	syl 12	. . . . . . . . 9 |- ((F C_ No /\ n e. F) -> (n` ( bday ` n)) = (/))
28	27	neeq1d 2028	. . . . . . . 8 |- ((F C_ No /\ n e. F) -> ((n` ( bday ` n)) =/= X <-> (/) =/= X))
29	25, 28	mpbiri 211	. . . . . . 7 |- ((F C_ No /\ n e. F) -> (n` ( bday ` n)) =/= X)
30		fveq2 4681	. . . . . . . . 9 |- (b = ( bday ` n) -> (n` b) = (n` ( bday ` n)))
31	30	neeq1d 2028	. . . . . . . 8 |- (b = ( bday ` n) -> ((n` b) =/= X <-> (n` ( bday ` n)) =/= X))
32	31	rcla4ev 2381	. . . . . . 7 |- ((( bday ` n) e. suc U.( bday "F) /\ (n` ( bday ` n)) =/= X) -> E.b e. suc U.( bday "F)(n` b) =/= X)
33	23, 29, 32	syl11anc 524	. . . . . 6 |- ((F C_ No /\ n e. F) -> E.b e. suc U.( bday "F)(n` b) =/= X)
34	33	r19.21aiva 2176	. . . . 5 |- (F C_ No -> A.n e. F E.b e. suc U.( bday "F)(n` b) =/= X)
35	3, 4, 34	syl2an 503	. . . 4 |- ((F e. A /\ F C_ No ) -> E.a e. On A.n e. F E.b e. a (n` b) =/= X)
36	35	ancoms 484	. . 3 |- ((F C_ No /\ F e. A) -> E.a e. On A.n e. F E.b e. a (n` b) =/= X)
37		onintrab2 3883	. . 3 |- (E.a e. On A.n e. F E.b e. a (n` b) =/= X <-> |^|{a e. On | A.n e. F E.b e. a (n` b) =/= X} e. On)
38	36, 37	sylib 215	. 2 |- ((F C_ No /\ F e. A) -> |^|{a e. On | A.n e. F E.b e. a (n` b) =/= X} e. On)
39		axfelem1.2	. 2 |- C = |^|{a e. On | A.n e. F E.b e. a (n` b) =/= X}
40	38, 39	syl5eqel 1975	1 |- ((F C_ No /\ F e. A) -> C e. On)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   C_ wss 2593  (/)c0 2875  {cpr 3045  U.cuni 3177  |^|cint 3214  Ord word 3656  Oncon0 3657  suc csuc 3659  dom cdm 3986  ran crn 3987  "cima 3989  Fun wfun 3992  ` cfv 3998  1oc1o 5172  2oc2o 5173   No csur 13981   bday cbday 13983

This theorem is referenced by:  axfelem2 14032  axfelem8 14038  axfelem9 14039

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-rab 2112  df-v 2294  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-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-fo 4012  df-fv 4014  df-mpt 5006  df-1o 5177  df-2o 5178  df-no 13984  df-bday 13986

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