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Mirrors > Home > MPE Home > Th. List > ttukeyg | Structured version Visualization version GIF version |
Description: The TeichmΓΌller-Tukey Lemma ttukey 10512 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 4341 | . . 3 β’ (π΄ β β β βπ§ π§ β π΄) | |
2 | ttukey2g 10510 | . . . . . 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 484 | . . . . . . 7 β’ ((π§ β π₯ β§ βπ¦ β π΄ Β¬ π₯ β π¦) β βπ¦ β π΄ Β¬ π₯ β π¦) | |
4 | 3 | reximi 3078 | . . . . . 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 395 β§ w3a 1084 βwal 1531 βwex 1773 β wcel 2098 β wne 2934 βwral 3055 βwrex 3064 β© cin 3942 β wss 3943 β wpss 3944 β c0 4317 π« cpw 4597 βͺ cuni 4902 dom cdm 5669 Fincfn 8938 cardccrd 9929 |
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-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-se 5625 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-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-fin 8942 df-card 9933 |
This theorem is referenced by: ttukey 10512 |
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