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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aomclem7 | Structured version Visualization version GIF version |
Description: Lemma for dfac11 43052. (𝑅1‘𝐴) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Ref | Expression |
---|---|
aomclem6.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)))} |
aomclem6.c | ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) |
aomclem6.d | ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) |
aomclem6.e | ⊢ 𝐸 = {〈𝑎, 𝑏〉 ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} |
aomclem6.f | ⊢ 𝐹 = {〈𝑎, 𝑏〉 ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} |
aomclem6.g | ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) |
aomclem6.h | ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺)) |
aomclem6.a | ⊢ (𝜑 → 𝐴 ∈ On) |
aomclem6.y | ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) |
Ref | Expression |
---|---|
aomclem7 | ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aomclem6.b | . . 3 ⊢ 𝐵 = {〈𝑎, 𝑏〉 ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} | |
2 | aomclem6.c | . . 3 ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) | |
3 | aomclem6.d | . . 3 ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) | |
4 | aomclem6.e | . . 3 ⊢ 𝐸 = {〈𝑎, 𝑏〉 ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} | |
5 | aomclem6.f | . . 3 ⊢ 𝐹 = {〈𝑎, 𝑏〉 ∣ ((rank‘𝑎) E (rank‘𝑏) ∨ ((rank‘𝑎) = (rank‘𝑏) ∧ 𝑎(𝑧‘suc (rank‘𝑎))𝑏))} | |
6 | aomclem6.g | . . 3 ⊢ 𝐺 = (if(dom 𝑧 = ∪ dom 𝑧, 𝐹, 𝐸) ∩ ((𝑅1‘dom 𝑧) × (𝑅1‘dom 𝑧))) | |
7 | aomclem6.h | . . 3 ⊢ 𝐻 = recs((𝑧 ∈ V ↦ 𝐺)) | |
8 | aomclem6.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ On) | |
9 | aomclem6.y | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | aomclem6 43049 | . 2 ⊢ (𝜑 → (𝐻‘𝐴) We (𝑅1‘𝐴)) |
11 | fvex 6917 | . . 3 ⊢ (𝐻‘𝐴) ∈ V | |
12 | weeq1 5670 | . . 3 ⊢ (𝑏 = (𝐻‘𝐴) → (𝑏 We (𝑅1‘𝐴) ↔ (𝐻‘𝐴) We (𝑅1‘𝐴))) | |
13 | 11, 12 | spcev 3605 | . 2 ⊢ ((𝐻‘𝐴) We (𝑅1‘𝐴) → ∃𝑏 𝑏 We (𝑅1‘𝐴)) |
14 | 10, 13 | syl 17 | 1 ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ∖ cdif 3947 ∩ cin 3949 ∅c0 4332 ifcif 4524 𝒫 cpw 4598 {csn 4624 ∪ cuni 4905 ∩ cint 4944 class class class wbr 5141 {copab 5203 ↦ cmpt 5223 E cep 5581 We wwe 5634 × cxp 5681 ◡ccnv 5682 dom cdm 5683 ran crn 5684 “ cima 5686 Oncon0 6382 suc csuc 6384 ‘cfv 6559 recscrecs 8406 Fincfn 8981 supcsup 9476 𝑅1cr1 9798 rankcrnk 9799 |
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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 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-rmo 3379 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 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-map 8864 df-en 8982 df-fin 8985 df-sup 9478 df-r1 9800 df-rank 9801 |
This theorem is referenced by: aomclem8 43051 |
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