![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > aomclem7 | Structured version Visualization version GIF version |
Description: Lemma for dfac11 42517. (π 1βπ΄) is well-orderable. (Contributed by Stefan O'Rear, 20-Jan-2015.) |
Ref | Expression |
---|---|
aomclem6.b | β’ π΅ = {β¨π, πβ© β£ βπ β (π 1ββͺ dom π§)((π β π β§ Β¬ π β π) β§ βπ β (π 1ββͺ dom π§)(π(π§ββͺ dom π§)π β (π β π β π β π)))} |
aomclem6.c | β’ πΆ = (π β V β¦ sup((π¦βπ), (π 1βdom π§), π΅)) |
aomclem6.d | β’ π· = recs((π β V β¦ (πΆβ((π 1βdom π§) β ran π)))) |
aomclem6.e | β’ πΈ = {β¨π, πβ© β£ β© (β‘π· β {π}) β β© (β‘π· β {π})} |
aomclem6.f | β’ πΉ = {β¨π, πβ© β£ ((rankβπ) E (rankβπ) β¨ ((rankβπ) = (rankβπ) β§ π(π§βsuc (rankβπ))π))} |
aomclem6.g | β’ πΊ = (if(dom π§ = βͺ dom π§, πΉ, πΈ) β© ((π 1βdom π§) Γ (π 1βdom π§))) |
aomclem6.h | β’ π» = recs((π§ β V β¦ πΊ)) |
aomclem6.a | β’ (π β π΄ β On) |
aomclem6.y | β’ (π β βπ β π« (π 1βπ΄)(π β β β (π¦βπ) β ((π« π β© Fin) β {β }))) |
Ref | Expression |
---|---|
aomclem7 | β’ (π β βπ π We (π 1βπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aomclem6.b | . . 3 β’ π΅ = {β¨π, πβ© β£ βπ β (π 1ββͺ dom π§)((π β π β§ Β¬ π β π) β§ βπ β (π 1ββͺ dom π§)(π(π§ββͺ dom π§)π β (π β π β π β π)))} | |
2 | aomclem6.c | . . 3 β’ πΆ = (π β V β¦ sup((π¦βπ), (π 1βdom π§), π΅)) | |
3 | aomclem6.d | . . 3 β’ π· = recs((π β V β¦ (πΆβ((π 1βdom π§) β ran π)))) | |
4 | aomclem6.e | . . 3 β’ πΈ = {β¨π, πβ© β£ β© (β‘π· β {π}) β β© (β‘π· β {π})} | |
5 | aomclem6.f | . . 3 β’ πΉ = {β¨π, πβ© β£ ((rankβπ) E (rankβπ) β¨ ((rankβπ) = (rankβπ) β§ π(π§βsuc (rankβπ))π))} | |
6 | aomclem6.g | . . 3 β’ πΊ = (if(dom π§ = βͺ dom π§, πΉ, πΈ) β© ((π 1βdom π§) Γ (π 1βdom π§))) | |
7 | aomclem6.h | . . 3 β’ π» = recs((π§ β V β¦ πΊ)) | |
8 | aomclem6.a | . . 3 β’ (π β π΄ β On) | |
9 | aomclem6.y | . . 3 β’ (π β βπ β π« (π 1βπ΄)(π β β β (π¦βπ) β ((π« π β© Fin) β {β }))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | aomclem6 42514 | . 2 β’ (π β (π»βπ΄) We (π 1βπ΄)) |
11 | fvex 6915 | . . 3 β’ (π»βπ΄) β V | |
12 | weeq1 5670 | . . 3 β’ (π = (π»βπ΄) β (π We (π 1βπ΄) β (π»βπ΄) We (π 1βπ΄))) | |
13 | 11, 12 | spcev 3595 | . 2 β’ ((π»βπ΄) We (π 1βπ΄) β βπ π We (π 1βπ΄)) |
14 | 10, 13 | syl 17 | 1 β’ (π β βπ π We (π 1βπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β¨ wo 845 = wceq 1533 βwex 1773 β wcel 2098 β wne 2937 βwral 3058 βwrex 3067 Vcvv 3473 β cdif 3946 β© cin 3948 β c0 4326 ifcif 4532 π« cpw 4606 {csn 4632 βͺ cuni 4912 β© cint 4953 class class class wbr 5152 {copab 5214 β¦ cmpt 5235 E cep 5585 We wwe 5636 Γ cxp 5680 β‘ccnv 5681 dom cdm 5682 ran crn 5683 β cima 5685 Oncon0 6374 suc csuc 6376 βcfv 6553 recscrecs 8397 Fincfn 8970 supcsup 9471 π 1cr1 9793 rankcrnk 9794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-map 8853 df-en 8971 df-fin 8974 df-sup 9473 df-r1 9795 df-rank 9796 |
This theorem is referenced by: aomclem8 42516 |
Copyright terms: Public domain | W3C validator |