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Theorem aomclem1 37450
Description: Lemma for dfac11 37458. 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 6199 . . 3 (𝑅1 dom 𝑧) ∈ V
2 vex 3201 . . . . . . . 8 𝑧 ∈ V
32dmex 7096 . . . . . . 7 dom 𝑧 ∈ V
43uniex 6950 . . . . . 6 dom 𝑧 ∈ V
54sucid 5802 . . . . 5 dom 𝑧 ∈ suc dom 𝑧
6 aomclem1.su . . . . 5 (𝜑 → dom 𝑧 = suc dom 𝑧)
75, 6syl5eleqr 2707 . . . 4 (𝜑 dom 𝑧 ∈ dom 𝑧)
8 aomclem1.we . . . 4 (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎))
9 fveq2 6189 . . . . . 6 (𝑎 = dom 𝑧 → (𝑧𝑎) = (𝑧 dom 𝑧))
10 fveq2 6189 . . . . . 6 (𝑎 = dom 𝑧 → (𝑅1𝑎) = (𝑅1 dom 𝑧))
119, 10weeq12d 37436 . . . . 5 (𝑎 = dom 𝑧 → ((𝑧𝑎) We (𝑅1𝑎) ↔ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)))
1211rspcva 3305 . . . 4 (( dom 𝑧 ∈ dom 𝑧 ∧ ∀𝑎 ∈ dom 𝑧(𝑧𝑎) We (𝑅1𝑎)) → (𝑧 dom 𝑧) We (𝑅1 dom 𝑧))
137, 8, 12syl2anc 693 . . 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 37439 . . 3 (((𝑅1 dom 𝑧) ∈ V ∧ (𝑧 dom 𝑧) We (𝑅1 dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1 dom 𝑧))
161, 13, 15sylancr 695 . 2 (𝜑𝐵 Or 𝒫 (𝑅1 dom 𝑧))
176fveq2d 6193 . . . 4 (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc dom 𝑧))
18 aomclem1.on . . . . 5 (𝜑 → dom 𝑧 ∈ On)
19 onuni 6990 . . . . 5 (dom 𝑧 ∈ On → dom 𝑧 ∈ On)
20 r1suc 8630 . . . . 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 2655 . . 3 (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1 dom 𝑧))
23 soeq2 5053 . . 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 384   = wceq 1482  wcel 1989  wral 2911  wrex 2912  Vcvv 3198  𝒫 cpw 4156   cuni 4434   class class class wbr 4651  {copab 4710   Or wor 5032   We wwe 5070  dom cdm 5112  Oncon0 5721  suc csuc 5723  cfv 5886  𝑅1cr1 8622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-2o 7558  df-map 7856  df-r1 8624
This theorem is referenced by:  aomclem2  37451
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