Theorem aomclem7 37630

Description: Lemma for dfac11 37632. ( R1 ` A ) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
aomclem6.b	|- B = { <. a , b >. | E. c e. ( R1 ` U. dom z ) ( ( c e. b /\ -. c e. a ) /\ A. d e. ( R1 ` U. dom z ) ( d ( z ` U. dom z ) c -> ( d e. a <-> d e. b ) ) ) }
aomclem6.c	|- C = ( a e. _V |-> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) )
aomclem6.d	|- D = recs ( ( a e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran a ) ) ) )
aomclem6.e	|- E = { <. a , b >. | |^| ( `' D " { a } ) e. |^| ( `' D " { b } ) }
aomclem6.f	|- F = { <. a , b >. | ( ( rank ` a ) _E ( rank ` b ) \/ ( ( rank ` a ) = ( rank ` b ) /\ a ( z ` suc ( rank ` a ) ) b ) ) }
aomclem6.g	|- G = ( if ( dom z = U. dom z , F , E ) i^i ( ( R1 ` dom z ) X. ( R1 ` dom z ) ) )
aomclem6.h	|- H = recs ( ( z e. _V |-> G ) )
aomclem6.a	|- ( ph -> A e. On )
aomclem6.y	|- ( ph -> A. a e. ~P ( R1 ` A ) ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) )

Assertion

Ref	Expression
aomclem7	|- ( ph -> E. b b We ( R1 ` A ) )

Distinct variable groups:   y, z, a, b, c, d   ph, a, b, c, d, z   C, a, b, c, d   D, a, b, c, d   A, a, b, c, d, z   H, a, b, c, d, z   G, d

Allowed substitution hints:   ph( y)   A( y)   B( y, z, a, b, c, d)   C( y, z)   D( y, z)   E( y, z, a, b, c, d)   F( y, z, a, b, c, d)   G( y, z, a, b, c)   H( y)

Proof of Theorem aomclem7

Step	Hyp	Ref	Expression
1		aomclem6.b	. . 3 |- B = { <. a , b >. | E. c e. ( R1 ` U. dom z ) ( ( c e. b /\ -. c e. a ) /\ A. d e. ( R1 ` U. dom z ) ( d ( z ` U. dom z ) c -> ( d e. a <-> d e. b ) ) ) }
2		aomclem6.c	. . 3 |- C = ( a e. _V |-> sup ( ( y ` a ) , ( R1 ` dom z ) , B ) )
3		aomclem6.d	. . 3 |- D = recs ( ( a e. _V |-> ( C ` ( ( R1 ` dom z ) \ ran a ) ) ) )
4		aomclem6.e	. . 3 |- E = { <. a , b >. | |^| ( `' D " { a } ) e. |^| ( `' D " { b } ) }
5		aomclem6.f	. . 3 |- F = { <. a , b >. | ( ( rank ` a ) _E ( rank ` b ) \/ ( ( rank ` a ) = ( rank ` b ) /\ a ( z ` suc ( rank ` a ) ) b ) ) }
6		aomclem6.g	. . 3 |- G = ( if ( dom z = U. dom z , F , E ) i^i ( ( R1 ` dom z ) X. ( R1 ` dom z ) ) )
7		aomclem6.h	. . 3 |- H = recs ( ( z e. _V |-> G ) )
8		aomclem6.a	. . 3 |- ( ph -> A e. On )
9		aomclem6.y	. . 3 |- ( ph -> A. a e. ~P ( R1 ` A ) ( a =/= (/) -> ( y ` a ) e. ( ( ~P a i^i Fin ) \ { (/) } ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	aomclem6 37629	. 2 |- ( ph -> ( H ` A ) We ( R1 ` A ) )
11		fvex 6201	. . 3 |- ( H ` A ) e. _V
12		weeq1 5102	. . 3 |- ( b = ( H ` A ) -> ( b We ( R1 ` A ) <-> ( H ` A ) We ( R1 ` A ) ) )
13	11, 12	spcev 3300	. 2 |- ( ( H ` A ) We ( R1 ` A ) -> E. b b We ( R1 ` A ) )
14	10, 13	syl 17	1 |- ( ph -> E. b b We ( R1 ` A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   i^i cin 3573   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   U.cuni 4436   |^|cint 4475   class class class wbr 4653   {copab 4712   |-> cmpt 4729   _E cep 5028   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Oncon0 5723   suc csuc 5725   ` cfv 5888  recscrecs 7467   Fincfn 7955   supcsup 8346   R1cr1 8625   rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-sup 8348  df-r1 8627  df-rank 8628

This theorem is referenced by:  aomclem8  37631