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Theorem aomclem1 43043
Description: Lemma for dfac11 43051. 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 6920 . . 3 (𝑅1 dom 𝑧) ∈ V
2 vex 3482 . . . . . . . 8 𝑧 ∈ V
32dmex 7932 . . . . . . 7 dom 𝑧 ∈ V
43uniex 7760 . . . . . 6 dom 𝑧 ∈ V
54sucid 6468 . . . . 5 dom 𝑧 ∈ suc dom 𝑧
6 aomclem1.su . . . . 5 (𝜑 → dom 𝑧 = suc dom 𝑧)
75, 6eleqtrrid 2846 . . . 4 (𝜑 dom 𝑧 ∈ dom 𝑧)
8 aomclem1.we . . . 4 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
9 fveq2 6907 . . . . . 6 (𝑎 = dom 𝑧 → (𝑧𝑎) = (𝑧 dom 𝑧))
10 fveq2 6907 . . . . . 6 (𝑎 = dom 𝑧 → (𝑅1𝑎) = (𝑅1 dom 𝑧))
119, 10weeq12d 5678 . . . . 5 (𝑎 = dom 𝑧 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)))
1211rspcva 3620 . . . 4 (( dom 𝑧 ∈ dom 𝑧 ∧ ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎)) → (𝑧 dom 𝑧) We (𝑅1 dom 𝑧))
137, 8, 12syl2anc 584 . . 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 43032 . . 3 (((𝑅1 dom 𝑧) ∈ V ∧ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1 dom 𝑧))
161, 13, 15sylancr 587 . 2 (𝜑𝐵 Or 𝒫 (𝑅1 dom 𝑧))
176fveq2d 6911 . . . 4 (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc dom 𝑧))
18 aomclem1.on . . . . 5 (𝜑 → dom 𝑧 ∈ On)
19 onuni 7808 . . . . 5 (dom 𝑧 ∈ On → dom 𝑧 ∈ On)
20 r1suc 9808 . . . . 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 2775 . . 3 (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1 dom 𝑧))
23 soeq2 5619 . . 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  {copab 5210   Or wor 5596   We wwe 5640  dom cdm 5689  Oncon0 6386  suc csuc 6388  cfv 6563  𝑅1cr1 9800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-map 8867  df-r1 9802
This theorem is referenced by:  aomclem2  43044
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