Theorem aomclem1 43898

Description: Lemma for dfac11 43906. 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 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.)

Hypotheses

Ref	Expression
aomclem1.b	⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
aomclem1.on	⊢ (𝜑 → dom 𝑧 ∈ On)
aomclem1.su	⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)
aomclem1.we	⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))

Assertion

Ref	Expression
aomclem1	⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧))

Distinct variable group:   𝑧,𝑎,𝑏,𝑐,𝑑

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

Proof of Theorem aomclem1

Step	Hyp	Ref	Expression
1		fvex 6892	. . 3 ⊢ (𝑅1‘∪ dom 𝑧) ∈ V
2		vex 3454	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	2	dmex 7907	. . . . . . 7 ⊢ dom 𝑧 ∈ V
4	3	uniex 7744	. . . . . 6 ⊢ ∪ dom 𝑧 ∈ V
5	4	sucid 6442	. . . . 5 ⊢ ∪ dom 𝑧 ∈ suc ∪ dom 𝑧
6		aomclem1.su	. . . . 5 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)
7	5, 6	eleqtrrid 2867	. . . 4 ⊢ (𝜑 → ∪ dom 𝑧 ∈ dom 𝑧)
8		aomclem1.we	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
9		fveq2 6879	. . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑧‘𝑎) = (𝑧‘∪ dom 𝑧))
10		fveq2 6879	. . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑅1‘𝑎) = (𝑅1‘∪ dom 𝑧))
11	9, 10	weeq12d 5644	. . . . 5 ⊢ (𝑎 = ∪ dom 𝑧 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)))
12	11	rspcva 3574	. . . 4 ⊢ ((∪ dom 𝑧 ∈ dom 𝑧 ∧ ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) → (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧))
13	7, 8, 12	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧))
14		aomclem1.b	. . . 4 ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))}
15	14	wepwso 43887	. . 3 ⊢ (((𝑅1‘∪ dom 𝑧) ∈ V ∧ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧))
16	1, 13, 15	sylancr 599	. 2 ⊢ (𝜑 → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧))
17	6	fveq2d 6883	. . . 4 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc ∪ dom 𝑧))
18		aomclem1.on	. . . . 5 ⊢ (𝜑 → dom 𝑧 ∈ On)
19		onuni 7788	. . . . 5 ⊢ (dom 𝑧 ∈ On → ∪ dom 𝑧 ∈ On)
20		r1suc 9755	. . . . 5 ⊢ (∪ dom 𝑧 ∈ On → (𝑅1‘suc ∪ dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
21	18, 19, 20	3syl 19	. . . 4 ⊢ (𝜑 → (𝑅1‘suc ∪ dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
22	17, 21	eqtrd 2795	. . 3 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
23		soeq2 5585	. . 3 ⊢ ((𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧) → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)))
24	22, 23	syl 18	. 2 ⊢ (𝜑 → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)))
25	16, 24	mpbird 260	1 ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  Vcvv 3450  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103  {copab 5167   Or wor 5562   We wwe 5607  dom cdm 5655  Oncon0 6357  suc csuc 6359  ‘cfv 6533  𝑅₁cr1 9747

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-map 8831  df-r1 9749

This theorem is used by:  aomclem2  43899