![]() |
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 41418. (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 41415 | . 2 β’ (π β (π»βπ΄) We (π 1βπ΄)) |
11 | fvex 6860 | . . 3 β’ (π»βπ΄) β V | |
12 | weeq1 5626 | . . 3 β’ (π = (π»βπ΄) β (π We (π 1βπ΄) β (π»βπ΄) We (π 1βπ΄))) | |
13 | 11, 12 | spcev 3568 | . 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 397 β¨ wo 846 = wceq 1542 βwex 1782 β wcel 2107 β wne 2944 βwral 3065 βwrex 3074 Vcvv 3448 β cdif 3912 β© cin 3914 β c0 4287 ifcif 4491 π« cpw 4565 {csn 4591 βͺ cuni 4870 β© cint 4912 class class class wbr 5110 {copab 5172 β¦ cmpt 5193 E cep 5541 We wwe 5592 Γ cxp 5636 β‘ccnv 5637 dom cdm 5638 ran crn 5639 β cima 5641 Oncon0 6322 suc csuc 6324 βcfv 6501 recscrecs 8321 Fincfn 8890 supcsup 9383 π 1cr1 9705 rankcrnk 9706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-map 8774 df-en 8891 df-fin 8894 df-sup 9385 df-r1 9707 df-rank 9708 |
This theorem is referenced by: aomclem8 41417 |
Copyright terms: Public domain | W3C validator |