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Theorem aomclem1 38043
Description: Lemma for dfac11 38051. 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 indexes 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 6314 . . 3 (𝑅1 dom 𝑧) ∈ V
2 vex 3307 . . . . . . . 8 𝑧 ∈ V
32dmex 7216 . . . . . . 7 dom 𝑧 ∈ V
43uniex 7070 . . . . . 6 dom 𝑧 ∈ V
54sucid 5917 . . . . 5 dom 𝑧 ∈ suc dom 𝑧
6 aomclem1.su . . . . 5 (𝜑 → dom 𝑧 = suc dom 𝑧)
75, 6syl5eleqr 2810 . . . 4 (𝜑 dom 𝑧 ∈ dom 𝑧)
8 aomclem1.we . . . 4 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
9 fveq2 6304 . . . . . 6 (𝑎 = dom 𝑧 → (𝑧𝑎) = (𝑧 dom 𝑧))
10 fveq2 6304 . . . . . 6 (𝑎 = dom 𝑧 → (𝑅1𝑎) = (𝑅1 dom 𝑧))
119, 10weeq12d 38029 . . . . 5 (𝑎 = dom 𝑧 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)))
1211rspcva 3411 . . . 4 (( dom 𝑧 ∈ dom 𝑧 ∧ ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎)) → (𝑧 dom 𝑧) We (𝑅1 dom 𝑧))
137, 8, 12syl2anc 696 . . 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 38032 . . 3 (((𝑅1 dom 𝑧) ∈ V ∧ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1 dom 𝑧))
161, 13, 15sylancr 698 . 2 (𝜑𝐵 Or 𝒫 (𝑅1 dom 𝑧))
176fveq2d 6308 . . . 4 (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc dom 𝑧))
18 aomclem1.on . . . . 5 (𝜑 → dom 𝑧 ∈ On)
19 onuni 7110 . . . . 5 (dom 𝑧 ∈ On → dom 𝑧 ∈ On)
20 r1suc 8746 . . . . 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 2758 . . 3 (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1 dom 𝑧))
23 soeq2 5159 . . 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 247 1 (𝜑𝐵 Or (𝑅1‘dom 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wral 3014  wrex 3015  Vcvv 3304  𝒫 cpw 4266   cuni 4544   class class class wbr 4760  {copab 4820   Or wor 5138   We wwe 5176  dom cdm 5218  Oncon0 5836  suc csuc 5838  cfv 6001  𝑅1cr1 8738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-2o 7681  df-map 7976  df-r1 8740
This theorem is referenced by:  aomclem2  38044
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