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Theorem aomclem1 41410
Description: Lemma for dfac11 41418. 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 6860 . . 3 (𝑅1 dom 𝑧) ∈ V
2 vex 3452 . . . . . . . 8 𝑧 ∈ V
32dmex 7853 . . . . . . 7 dom 𝑧 ∈ V
43uniex 7683 . . . . . 6 dom 𝑧 ∈ V
54sucid 6404 . . . . 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 6847 . . . . . 6 (𝑎 = dom 𝑧 → (𝑧𝑎) = (𝑧 dom 𝑧))
10 fveq2 6847 . . . . . 6 (𝑎 = dom 𝑧 → (𝑅1𝑎) = (𝑅1 dom 𝑧))
119, 10weeq12d 41396 . . . . 5 (𝑎 = dom 𝑧 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)))
1211rspcva 3582 . . . 4 (( dom 𝑧 ∈ dom 𝑧 ∧ ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎)) → (𝑧 dom 𝑧) We (𝑅1 dom 𝑧))
137, 8, 12syl2anc 585 . . 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 41399 . . 3 (((𝑅1 dom 𝑧) ∈ V ∧ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1 dom 𝑧))
161, 13, 15sylancr 588 . 2 (𝜑𝐵 Or 𝒫 (𝑅1 dom 𝑧))
176fveq2d 6851 . . . 4 (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc dom 𝑧))
18 aomclem1.on . . . . 5 (𝜑 → dom 𝑧 ∈ On)
19 onuni 7728 . . . . 5 (dom 𝑧 ∈ On → dom 𝑧 ∈ On)
20 r1suc 9713 . . . . 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 5572 . . 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 257 1 (𝜑𝐵 Or (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  wrex 3074  Vcvv 3448  𝒫 cpw 4565   cuni 4870   class class class wbr 5110  {copab 5172   Or wor 5549   We wwe 5592  dom cdm 5638  Oncon0 6322  suc csuc 6324  cfv 6501  𝑅1cr1 9705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-map 8774  df-r1 9707
This theorem is referenced by:  aomclem2  41411
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