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| Mirrors > Home > MPE Home > Th. List > tskinf | Structured version Visualization version GIF version | ||
| Description: A nonempty Tarski class is infinite. (Contributed by FL, 22-Feb-2011.) |
| Ref | Expression |
|---|---|
| tskinf | β’ ((π β Tarski β§ π β β ) β Ο βΌ π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | r111 9766 | . . . 4 β’ π 1:Onβ1-1βV | |
| 2 | omsson 7855 | . . . 4 β’ Ο β On | |
| 3 | omex 9634 | . . . . 5 β’ Ο β V | |
| 4 | 3 | f1imaen 9008 | . . . 4 β’ ((π 1:Onβ1-1βV β§ Ο β On) β (π 1 β Ο) β Ο) |
| 5 | 1, 2, 4 | mp2an 690 | . . 3 β’ (π 1 β Ο) β Ο |
| 6 | 5 | ensymi 8996 | . 2 β’ Ο β (π 1 β Ο) |
| 7 | simpl 483 | . . 3 β’ ((π β Tarski β§ π β β ) β π β Tarski) | |
| 8 | tskr1om 10758 | . . 3 β’ ((π β Tarski β§ π β β ) β (π 1 β Ο) β π) | |
| 9 | ssdomg 8992 | . . 3 β’ (π β Tarski β ((π 1 β Ο) β π β (π 1 β Ο) βΌ π)) | |
| 10 | 7, 8, 9 | sylc 65 | . 2 β’ ((π β Tarski β§ π β β ) β (π 1 β Ο) βΌ π) |
| 11 | endomtr 9004 | . 2 β’ ((Ο β (π 1 β Ο) β§ (π 1 β Ο) βΌ π) β Ο βΌ π) | |
| 12 | 6, 10, 11 | sylancr 587 | 1 β’ ((π β Tarski β§ π β β ) β Ο βΌ π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 396 β wcel 2106 β wne 2940 Vcvv 3474 β wss 3947 β c0 4321 class class class wbr 5147 β cima 5678 Oncon0 6361 β1-1βwf1 6537 Οcom 7851 β cen 8932 βΌ cdom 8933 π 1cr1 9753 Tarskictsk 10739 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-r1 9755 df-tsk 10740 |
| This theorem is referenced by: tskpr 10761 |
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