![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > aomclem1 | Structured version Visualization version GIF version |
Description: Lemma for dfac11 42106. 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 6903 | . . 3 ⊢ (𝑅1‘∪ dom 𝑧) ∈ V | |
2 | vex 3476 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 2 | dmex 7904 | . . . . . . 7 ⊢ dom 𝑧 ∈ V |
4 | 3 | uniex 7733 | . . . . . 6 ⊢ ∪ dom 𝑧 ∈ V |
5 | 4 | sucid 6445 | . . . . 5 ⊢ ∪ dom 𝑧 ∈ suc ∪ dom 𝑧 |
6 | aomclem1.su | . . . . 5 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) | |
7 | 5, 6 | eleqtrrid 2838 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑧 ∈ dom 𝑧) |
8 | aomclem1.we | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) | |
9 | fveq2 6890 | . . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑧‘𝑎) = (𝑧‘∪ dom 𝑧)) | |
10 | fveq2 6890 | . . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑅1‘𝑎) = (𝑅1‘∪ dom 𝑧)) | |
11 | 9, 10 | weeq12d 42084 | . . . . 5 ⊢ (𝑎 = ∪ dom 𝑧 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧))) |
12 | 11 | rspcva 3609 | . . . 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 582 | . . 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 42087 | . . 3 ⊢ (((𝑅1‘∪ dom 𝑧) ∈ V ∧ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧)) → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)) |
16 | 1, 13, 15 | sylancr 585 | . 2 ⊢ (𝜑 → 𝐵 Or 𝒫 (𝑅1‘∪ dom 𝑧)) |
17 | 6 | fveq2d 6894 | . . . 4 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc ∪ dom 𝑧)) |
18 | aomclem1.on | . . . . 5 ⊢ (𝜑 → dom 𝑧 ∈ On) | |
19 | onuni 7778 | . . . . 5 ⊢ (dom 𝑧 ∈ On → ∪ dom 𝑧 ∈ On) | |
20 | r1suc 9767 | . . . . 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 2770 | . . 3 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧)) |
23 | soeq2 5609 | . . 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 256 | 1 ⊢ (𝜑 → 𝐵 Or (𝑅1‘dom 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 Vcvv 3472 𝒫 cpw 4601 ∪ cuni 4907 class class class wbr 5147 {copab 5209 Or wor 5586 We wwe 5629 dom cdm 5675 Oncon0 6363 suc csuc 6365 ‘cfv 6542 𝑅1cr1 9759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-map 8824 df-r1 9761 |
This theorem is referenced by: aomclem2 42099 |
Copyright terms: Public domain | W3C validator |