Theorem aomclem1 43029

Description: Lemma for dfac11 43037. 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 6899	. . 3 ⊢ (𝑅1‘∪ dom 𝑧) ∈ V
2		vex 3467	. . . . . . . 8 ⊢ 𝑧 ∈ V
3	2	dmex 7913	. . . . . . 7 ⊢ dom 𝑧 ∈ V
4	3	uniex 7743	. . . . . 6 ⊢ ∪ dom 𝑧 ∈ V
5	4	sucid 6446	. . . . 5 ⊢ ∪ dom 𝑧 ∈ suc ∪ dom 𝑧
6		aomclem1.su	. . . . 5 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧)
7	5, 6	eleqtrrid 2840	. . . 4 ⊢ (𝜑 → ∪ dom 𝑧 ∈ dom 𝑧)
8		aomclem1.we	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎))
9		fveq2 6886	. . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑧‘𝑎) = (𝑧‘∪ dom 𝑧))
10		fveq2 6886	. . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑅1‘𝑎) = (𝑅1‘∪ dom 𝑧))
11	9, 10	weeq12d 5654	. . . . 5 ⊢ (𝑎 = ∪ dom 𝑧 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)))
12	11	rspcva 3603	. . . 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 584	. . 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 43018	. . 3 ⊢ (((𝑅1‘∪ dom 𝑧) ∈ V ∧ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧))
16	1, 13, 15	sylancr 587	. 2 ⊢ (𝜑 → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧))
17	6	fveq2d 6890	. . . 4 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc ∪ dom 𝑧))
18		aomclem1.on	. . . . 5 ⊢ (𝜑 → dom 𝑧 ∈ On)
19		onuni 7790	. . . . 5 ⊢ (dom 𝑧 ∈ On → ∪ dom 𝑧 ∈ On)
20		r1suc 9792	. . . . 5 ⊢ (∪ dom 𝑧 ∈ On → (𝑅1‘suc ∪ dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
21	18, 19, 20	3syl 18	. . . 4 ⊢ (𝜑 → (𝑅1‘suc ∪ dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
22	17, 21	eqtrd 2769	. . 3 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧))
23		soeq2 5594	. . 3 ⊢ ((𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧) → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)))
24	22, 23	syl 17	. 2 ⊢ (𝜑 → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)))
25	16, 24	mpbird 257	1 ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  Vcvv 3463  𝒫 cpw 4580  ∪ cuni 4887   class class class wbr 5123  {copab 5185   Or wor 5571   We wwe 5616  dom cdm 5665  Oncon0 6363  suc csuc 6365  ‘cfv 6541  𝑅₁cr1 9784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-map 8850  df-r1 9786

This theorem is referenced by:  aomclem2  43030