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| Description: Lemma for dfac11 43079.  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 (𝑅1‘𝐴).  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.) | 
| Ref | Expression | 
|---|---|
| aomclem1.b | ⊢ 𝐵 = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} | 
| aomclem1.on | ⊢ (𝜑 → dom 𝑧 ∈ On) | 
| aomclem1.su | ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) | 
| aomclem1.we | ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) | 
| Ref | Expression | 
|---|---|
| aomclem1 | ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvex 6918 | . . 3 ⊢ (𝑅1‘∪ dom 𝑧) ∈ V | |
| 2 | vex 3483 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
| 3 | 2 | dmex 7932 | . . . . . . 7 ⊢ dom 𝑧 ∈ V | 
| 4 | 3 | uniex 7762 | . . . . . 6 ⊢ ∪ dom 𝑧 ∈ V | 
| 5 | 4 | sucid 6465 | . . . . 5 ⊢ ∪ dom 𝑧 ∈ suc ∪ dom 𝑧 | 
| 6 | aomclem1.su | . . . . 5 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) | |
| 7 | 5, 6 | eleqtrrid 2847 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑧 ∈ dom 𝑧) | 
| 8 | aomclem1.we | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) | |
| 9 | fveq2 6905 | . . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑧‘𝑎) = (𝑧‘∪ dom 𝑧)) | |
| 10 | fveq2 6905 | . . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑅1‘𝑎) = (𝑅1‘∪ dom 𝑧)) | |
| 11 | 9, 10 | weeq12d 5673 | . . . . 5 ⊢ (𝑎 = ∪ dom 𝑧 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧))) | 
| 12 | 11 | rspcva 3619 | . . . 4 ⊢ ((∪ dom 𝑧 ∈ dom 𝑧 ∧ ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) → (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)) | 
| 13 | 7, 8, 12 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)) | 
| 14 | aomclem1.b | . . . 4 ⊢ 𝐵 = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} | |
| 15 | 14 | wepwso 43060 | . . 3 ⊢ (((𝑅1‘∪ dom 𝑧) ∈ V ∧ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)) | 
| 16 | 1, 13, 15 | sylancr 587 | . 2 ⊢ (𝜑 → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)) | 
| 17 | 6 | fveq2d 6909 | . . . 4 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc ∪ dom 𝑧)) | 
| 18 | aomclem1.on | . . . . 5 ⊢ (𝜑 → dom 𝑧 ∈ On) | |
| 19 | onuni 7809 | . . . . 5 ⊢ (dom 𝑧 ∈ On → ∪ dom 𝑧 ∈ On) | |
| 20 | r1suc 9811 | . . . . 5 ⊢ (∪ dom 𝑧 ∈ On → (𝑅1‘suc ∪ dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧)) | |
| 21 | 18, 19, 20 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑅1‘suc ∪ dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧)) | 
| 22 | 17, 21 | eqtrd 2776 | . . 3 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧)) | 
| 23 | soeq2 5613 | . . 3 ⊢ ((𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧) → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧))) | |
| 24 | 22, 23 | syl 17 | . 2 ⊢ (𝜑 → (𝐵 Or (𝑅1‘dom 𝑧) ↔ 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧))) | 
| 25 | 16, 24 | mpbird 257 | 1 ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 Vcvv 3479 𝒫 cpw 4599 ∪ cuni 4906 class class class wbr 5142 {copab 5204 Or wor 5590 We wwe 5635 dom cdm 5684 Oncon0 6383 suc csuc 6385 ‘cfv 6560 𝑅1cr1 9803 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-map 8869 df-r1 9805 | 
| This theorem is referenced by: aomclem2 43072 | 
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