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Mirrors > Home > MPE Home > Th. List > ttukeyg | Structured version Visualization version GIF version |
Description: The TeichmΓΌller-Tukey Lemma ttukey 10549 stated with the "choice" as an antecedent (the hypothesis βͺ π΄ β dom card says that βͺ π΄ is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
ttukeyg | β’ ((βͺ π΄ β dom card β§ π΄ β β β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4350 | . . 3 β’ (π΄ β β β βπ§ π§ β π΄) | |
2 | ttukey2g 10547 | . . . . . 6 β’ ((βͺ π΄ β dom card β§ π§ β π΄ β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ (π§ β π₯ β§ βπ¦ β π΄ Β¬ π₯ β π¦)) | |
3 | simpr 483 | . . . . . . 7 β’ ((π§ β π₯ β§ βπ¦ β π΄ Β¬ π₯ β π¦) β βπ¦ β π΄ Β¬ π₯ β π¦) | |
4 | 3 | reximi 3081 | . . . . . 6 β’ (βπ₯ β π΄ (π§ β π₯ β§ βπ¦ β π΄ Β¬ π₯ β π¦) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) |
5 | 2, 4 | syl 17 | . . . . 5 β’ ((βͺ π΄ β dom card β§ π§ β π΄ β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) |
6 | 5 | 3exp 1116 | . . . 4 β’ (βͺ π΄ β dom card β (π§ β π΄ β (βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦))) |
7 | 6 | exlimdv 1928 | . . 3 β’ (βͺ π΄ β dom card β (βπ§ π§ β π΄ β (βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦))) |
8 | 1, 7 | biimtrid 241 | . 2 β’ (βͺ π΄ β dom card β (π΄ β β β (βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦))) |
9 | 8 | 3imp 1108 | 1 β’ ((βͺ π΄ β dom card β§ π΄ β β β§ βπ₯(π₯ β π΄ β (π« π₯ β© Fin) β π΄)) β βπ₯ β π΄ βπ¦ β π΄ Β¬ π₯ β π¦) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 394 β§ w3a 1084 βwal 1531 βwex 1773 β wcel 2098 β wne 2937 βwral 3058 βwrex 3067 β© cin 3948 β wss 3949 β wpss 3950 β c0 4326 π« cpw 4606 βͺ cuni 4912 dom cdm 5682 Fincfn 8970 cardccrd 9966 |
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-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-se 5638 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-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-fin 8974 df-card 9970 |
This theorem is referenced by: ttukey 10549 |
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