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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 42364. (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 42361 | . 2 β’ (π β (π»βπ΄) We (π 1βπ΄)) |
11 | fvex 6897 | . . 3 β’ (π»βπ΄) β V | |
12 | weeq1 5657 | . . 3 β’ (π = (π»βπ΄) β (π We (π 1βπ΄) β (π»βπ΄) We (π 1βπ΄))) | |
13 | 11, 12 | spcev 3590 | . 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 395 β¨ wo 844 = wceq 1533 βwex 1773 β wcel 2098 β wne 2934 βwral 3055 βwrex 3064 Vcvv 3468 β cdif 3940 β© cin 3942 β c0 4317 ifcif 4523 π« cpw 4597 {csn 4623 βͺ cuni 4902 β© cint 4943 class class class wbr 5141 {copab 5203 β¦ cmpt 5224 E cep 5572 We wwe 5623 Γ cxp 5667 β‘ccnv 5668 dom cdm 5669 ran crn 5670 β cima 5672 Oncon0 6357 suc csuc 6359 βcfv 6536 recscrecs 8368 Fincfn 8938 supcsup 9434 π 1cr1 9756 rankcrnk 9757 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-map 8821 df-en 8939 df-fin 8942 df-sup 9436 df-r1 9758 df-rank 9759 |
This theorem is referenced by: aomclem8 42363 |
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