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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 43051. 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 6920 | . . 3 ⊢ (𝑅1‘∪ dom 𝑧) ∈ V | |
2 | vex 3482 | . . . . . . . 8 ⊢ 𝑧 ∈ V | |
3 | 2 | dmex 7932 | . . . . . . 7 ⊢ dom 𝑧 ∈ V |
4 | 3 | uniex 7760 | . . . . . 6 ⊢ ∪ dom 𝑧 ∈ V |
5 | 4 | sucid 6468 | . . . . 5 ⊢ ∪ dom 𝑧 ∈ suc ∪ dom 𝑧 |
6 | aomclem1.su | . . . . 5 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) | |
7 | 5, 6 | eleqtrrid 2846 | . . . 4 ⊢ (𝜑 → ∪ dom 𝑧 ∈ dom 𝑧) |
8 | aomclem1.we | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) | |
9 | fveq2 6907 | . . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑧‘𝑎) = (𝑧‘∪ dom 𝑧)) | |
10 | fveq2 6907 | . . . . . 6 ⊢ (𝑎 = ∪ dom 𝑧 → (𝑅1‘𝑎) = (𝑅1‘∪ dom 𝑧)) | |
11 | 9, 10 | weeq12d 5678 | . . . . 5 ⊢ (𝑎 = ∪ dom 𝑧 → ((𝑧‘𝑎) We (𝑅1‘𝑎) ↔ (𝑧‘∪ dom 𝑧) We (𝑅1‘∪ dom 𝑧))) |
12 | 11 | rspcva 3620 | . . . 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 43032 | . . 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 6911 | . . . 4 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = (𝑅1‘suc ∪ dom 𝑧)) |
18 | aomclem1.on | . . . . 5 ⊢ (𝜑 → dom 𝑧 ∈ On) | |
19 | onuni 7808 | . . . . 5 ⊢ (dom 𝑧 ∈ On → ∪ dom 𝑧 ∈ On) | |
20 | r1suc 9808 | . . . . 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 2775 | . . 3 ⊢ (𝜑 → (𝑅1‘dom 𝑧) = 𝒫 (𝑅1‘∪ dom 𝑧)) |
23 | soeq2 5619 | . . 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 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 𝒫 cpw 4605 ∪ cuni 4912 class class class wbr 5148 {copab 5210 Or wor 5596 We wwe 5640 dom cdm 5689 Oncon0 6386 suc csuc 6388 ‘cfv 6563 𝑅1cr1 9800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-map 8867 df-r1 9802 |
This theorem is referenced by: aomclem2 43044 |
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