Theorem aomclem1 37624

Description: Lemma for dfac11 37632. This is the beginning of the proof that multiple choice is equivalent to choice. Our goal is to construct, by transfinite recursion, a well-ordering of ( R1 ` A ). In what follows, A is the index of the rank we wish to well-order, z is the collection of well-orderings constructed so far, dom z is the set of ordinal indexes of constructed ranks i.e. the next rank to construct, and y is a postulated multiple-choice function.

Successor case 1, define a simple ordering from the well-ordered predecessor. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypotheses

Ref	Expression
aomclem1.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 ) ) ) }
aomclem1.on	|- ( ph -> dom z e. On )
aomclem1.su	|- ( ph -> dom z = suc U. dom z )
aomclem1.we	|- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )

Assertion

Ref	Expression
aomclem1	|- ( ph -> B Or ( R1 ` dom z ) )

Distinct variable group:   z, a, b, c, d

Allowed substitution hints:   ph( z, a, b, c, d)   B( z, a, b, c, d)

Proof of Theorem aomclem1

Step	Hyp	Ref	Expression
1		fvex 6201	. . 3 |- ( R1 ` U. dom z ) e. _V
2		vex 3203	. . . . . . . 8 |- z e. _V
3	2	dmex 7099	. . . . . . 7 |- dom z e. _V
4	3	uniex 6953	. . . . . 6 |- U. dom z e. _V
5	4	sucid 5804	. . . . 5 |- U. dom z e. suc U. dom z
6		aomclem1.su	. . . . 5 |- ( ph -> dom z = suc U. dom z )
7	5, 6	syl5eleqr 2708	. . . 4 |- ( ph -> U. dom z e. dom z )
8		aomclem1.we	. . . 4 |- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )
9		fveq2 6191	. . . . . 6 |- ( a = U. dom z -> ( z ` a ) = ( z ` U. dom z ) )
10		fveq2 6191	. . . . . 6 |- ( a = U. dom z -> ( R1 ` a ) = ( R1 ` U. dom z ) )
11	9, 10	weeq12d 37610	. . . . 5 |- ( a = U. dom z -> ( ( z ` a ) We ( R1 ` a ) <-> ( z ` U. dom z ) We ( R1 ` U. dom z ) ) )
12	11	rspcva 3307	. . . 4 |- ( ( U. dom z e. dom z /\ A. a e. dom z ( z ` a ) We ( R1 ` a ) ) -> ( z ` U. dom z ) We ( R1 ` U. dom z ) )
13	7, 8, 12	syl2anc 693	. . 3 |- ( ph -> ( z ` U. dom z ) We ( R1 ` U. dom z ) )
14		aomclem1.b	. . . 4 |- 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 ) ) ) }
15	14	wepwso 37613	. . 3 |- ( ( ( R1 ` U. dom z ) e. _V /\ ( z ` U. dom z ) We ( R1 ` U. dom z ) ) -> B Or ~P ( R1 ` U. dom z ) )
16	1, 13, 15	sylancr 695	. 2 |- ( ph -> B Or ~P ( R1 ` U. dom z ) )
17	6	fveq2d 6195	. . . 4 |- ( ph -> ( R1 ` dom z ) = ( R1 ` suc U. dom z ) )
18		aomclem1.on	. . . . 5 |- ( ph -> dom z e. On )
19		onuni 6993	. . . . 5 |- ( dom z e. On -> U. dom z e. On )
20		r1suc 8633	. . . . 5 |- ( U. dom z e. On -> ( R1 ` suc U. dom z ) = ~P ( R1 ` U. dom z ) )
21	18, 19, 20	3syl 18	. . . 4 |- ( ph -> ( R1 ` suc U. dom z ) = ~P ( R1 ` U. dom z ) )
22	17, 21	eqtrd 2656	. . 3 |- ( ph -> ( R1 ` dom z ) = ~P ( R1 ` U. dom z ) )
23		soeq2 5055	. . 3 |- ( ( R1 ` dom z ) = ~P ( R1 ` U. dom z ) -> ( B Or ( R1 ` dom z ) <-> B Or ~P ( R1 ` U. dom z ) ) )
24	22, 23	syl 17	. 2 |- ( ph -> ( B Or ( R1 ` dom z ) <-> B Or ~P ( R1 ` U. dom z ) ) )
25	16, 24	mpbird 247	1 |- ( ph -> B Or ( R1 ` dom z ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   {copab 4712   Or wor 5034   We wwe 5072   dom cdm 5114   Oncon0 5723   suc csuc 5725   ` cfv 5888   R1cr1 8625

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-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-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-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-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-map 7859  df-r1 8627

This theorem is referenced by:  aomclem2  37625