Theorem aomclem1

Description: Lemma for dfac11 . 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 ( R1A ) . 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 indices 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)

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

Proof

Step	Hyp	Ref	Expression
1		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 ) ) ) }
2		aomclem1.on	|- ( ph -> dom z e. On )
3		aomclem1.su	|- ( ph -> dom z = suc U. dom z )
4		aomclem1.we	|- ( ph -> A. a e. dom z ( z ` a ) We ( R1 ` a ) )
5		fvex	|- ( R1 ` U. dom z ) e. _V
6		vex	|- z e. _V
7	6	dmex	|- dom z e. _V
8	7	uniex	|- U. dom z e. _V
9	8	sucid	|- U. dom z e. suc U. dom z
10	9 3	eleqtrrid	|- ( ph -> U. dom z e. dom z )
11		fveq2	|- ( a = U. dom z -> ( z ` a ) = ( z ` U. dom z ) )
12		fveq2	|- ( a = U. dom z -> ( R1 ` a ) = ( R1 ` U. dom z ) )
13	11 12	weeq12d	|- ( a = U. dom z -> ( ( z ` a ) We ( R1 ` a ) <-> ( z ` U. dom z ) We ( R1 ` U. dom z ) ) )
14	13	rspcva	|- ( ( 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 ) )
15	10 4 14	syl2anc	|- ( ph -> ( z ` U. dom z ) We ( R1 ` U. dom z ) )
16	1	wepwso	|- ( ( ( R1 ` U. dom z ) e. _V /\ ( z ` U. dom z ) We ( R1 ` U. dom z ) ) -> B Or ~P ( R1 ` U. dom z ) )
17	5 15 16	sylancr	|- ( ph -> B Or ~P ( R1 ` U. dom z ) )
18	3	fveq2d	|- ( ph -> ( R1 ` dom z ) = ( R1 ` suc U. dom z ) )
19		onuni	|- ( dom z e. On -> U. dom z e. On )
20		r1suc	|- ( U. dom z e. On -> ( R1 ` suc U. dom z ) = ~P ( R1 ` U. dom z ) )
21	2 19 20	3syl	|- ( ph -> ( R1 ` suc U. dom z ) = ~P ( R1 ` U. dom z ) )
22	18 21	eqtrd	|- ( ph -> ( R1 ` dom z ) = ~P ( R1 ` U. dom z ) )
23		soeq2	|- ( ( R1 ` dom z ) = ~P ( R1 ` U. dom z ) -> ( B Or ( R1 ` dom z ) <-> B Or ~P ( R1 ` U. dom z ) ) )
24	22 23	syl	|- ( ph -> ( B Or ( R1 ` dom z ) <-> B Or ~P ( R1 ` U. dom z ) ) )
25	17 24	mpbird	|- ( ph -> B Or ( R1 ` dom z ) )