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Theorem aomclem1 40582
Description: Lemma for dfac11 40590. 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.)

Hypotheses
Ref Expression
aomclem1.b 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
aomclem1.on (𝜑 → dom 𝑧 ∈ On)
aomclem1.su (𝜑 → dom 𝑧 = suc dom 𝑧)
aomclem1.we (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
Assertion
Ref Expression
aomclem1 (𝜑𝐵 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
StepHypRef Expression
1 fvex 6730 . . 3 (𝑅1 dom 𝑧) ∈ V
2 vex 3412 . . . . . . . 8 𝑧 ∈ V
32dmex 7689 . . . . . . 7 dom 𝑧 ∈ V
43uniex 7529 . . . . . 6 dom 𝑧 ∈ V
54sucid 6292 . . . . 5 dom 𝑧 ∈ suc dom 𝑧
6 aomclem1.su . . . . 5 (𝜑 → dom 𝑧 = suc dom 𝑧)
75, 6eleqtrrid 2845 . . . 4 (𝜑 dom 𝑧 ∈ dom 𝑧)
8 aomclem1.we . . . 4 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
9 fveq2 6717 . . . . . 6 (𝑎 = dom 𝑧 → (𝑧𝑎) = (𝑧 dom 𝑧))
10 fveq2 6717 . . . . . 6 (𝑎 = dom 𝑧 → (𝑅1𝑎) = (𝑅1 dom 𝑧))
119, 10weeq12d 40568 . . . . 5 (𝑎 = dom 𝑧 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)))
1211rspcva 3535 . . . 4 (( dom 𝑧 ∈ dom 𝑧 ∧ ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎)) → (𝑧 dom 𝑧) We (𝑅1 dom 𝑧))
137, 8, 12syl2anc 587 . . 3 (𝜑 → (𝑧 dom 𝑧) We (𝑅1 dom 𝑧))
14 aomclem1.b . . . 4 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1 dom 𝑧)((𝑐𝑏 ∧ ¬ 𝑐𝑎) ∧ ∀𝑑 ∈ (𝑅1 dom 𝑧)(𝑑(𝑧 dom 𝑧)𝑐 → (𝑑𝑎𝑑𝑏)))}
1514wepwso 40571 . . 3 (((𝑅1 dom 𝑧) ∈ V ∧ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1 dom 𝑧))
161, 13, 15sylancr 590 . 2 (𝜑𝐵 Or 𝒫 (𝑅1 dom 𝑧))
176fveq2d 6721 . . . 4 (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc dom 𝑧))
18 aomclem1.on . . . . 5 (𝜑 → dom 𝑧 ∈ On)
19 onuni 7572 . . . . 5 (dom 𝑧 ∈ On → dom 𝑧 ∈ On)
20 r1suc 9386 . . . . 5 ( dom 𝑧 ∈ On → (𝑅1‘suc dom 𝑧) = 𝒫 (𝑅1 dom 𝑧))
2118, 19, 203syl 18 . . . 4 (𝜑 → (𝑅1‘suc dom 𝑧) = 𝒫 (𝑅1 dom 𝑧))
2217, 21eqtrd 2777 . . 3 (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1 dom 𝑧))
23 soeq2 5490 . . 3 ((𝑅1‘dom 𝑧) = 𝒫 (𝑅1 dom 𝑧) → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1 dom 𝑧)))
2422, 23syl 17 . 2 (𝜑 → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1 dom 𝑧)))
2516, 24mpbird 260 1 (𝜑𝐵 Or (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  wrex 3062  Vcvv 3408  𝒫 cpw 4513   cuni 4819   class class class wbr 5053  {copab 5115   Or wor 5467   We wwe 5508  dom cdm 5551  Oncon0 6213  suc csuc 6215  cfv 6380  𝑅1cr1 9378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-map 8510  df-r1 9380
This theorem is referenced by:  aomclem2  40583
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