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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aomclem1 | Structured version Visualization version GIF version |
Description: Lemma for dfac11 42107. 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 6904 | . . 3 ⊢ (𝑅1‘∪ dom 𝑧) ∈ V | |
2 | vex 3477 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 2 | dmex 7905 | . . . . . . 7 ⊢ dom 𝑧 ∈ V |
4 | 3 | uniex 7734 | . . . . . 6 ⊢ ∪ dom 𝑧 ∈ V |
5 | 4 | sucid 6446 | . . . . 5 ⊢ ∪ dom 𝑧 ∈ suc ∪ dom 𝑧 |
6 | aomclem1.su | . . . . 5 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) | |
7 | 5, 6 | eleqtrrid 2839 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑧 ∈ dom 𝑧) |
8 | aomclem1.we | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) | |
9 | fveq2 6891 | . . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑧‘𝑎) = (𝑧‘∪ dom 𝑧)) | |
10 | fveq2 6891 | . . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑅1‘𝑎) = (𝑅1‘∪ dom 𝑧)) | |
11 | 9, 10 | weeq12d 42085 | . . . . 5 ⊢ (𝑎 = ∪ dom 𝑧 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧))) |
12 | 11 | rspcva 3610 | . . . 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 583 | . . 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 42088 | . . 3 ⊢ (((𝑅1‘∪ dom 𝑧) ∈ V ∧ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)) |
16 | 1, 13, 15 | sylancr 586 | . 2 ⊢ (𝜑 → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)) |
17 | 6 | fveq2d 6895 | . . . 4 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc ∪ dom 𝑧)) |
18 | aomclem1.on | . . . . 5 ⊢ (𝜑 → dom 𝑧 ∈ On) | |
19 | onuni 7779 | . . . . 5 ⊢ (dom 𝑧 ∈ On → ∪ dom 𝑧 ∈ On) | |
20 | r1suc 9768 | . . . . 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 2771 | . . 3 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧)) |
23 | soeq2 5610 | . . 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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 Vcvv 3473 𝒫 cpw 4602 ∪ cuni 4908 class class class wbr 5148 {copab 5210 Or wor 5587 We wwe 5630 dom cdm 5676 Oncon0 6364 suc csuc 6366 ‘cfv 6543 𝑅1cr1 9760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-map 8825 df-r1 9762 |
This theorem is referenced by: aomclem2 42100 |
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