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Mirrors > Home > MPE Home > Th. List > rankr1a | Structured version Visualization version GIF version |
Description: A relationship between rank and π 1, clearly equivalent to ssrankr1 9836 and friends through trichotomy, but in Raph's opinion considerably more intuitive. See rankr1b 9865 for the subset version. (Contributed by Raph Levien, 29-May-2004.) |
Ref | Expression |
---|---|
rankid.1 | β’ π΄ β V |
Ref | Expression |
---|---|
rankr1a | β’ (π΅ β On β (π΄ β (π 1βπ΅) β (rankβπ΄) β π΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankid.1 | . . . 4 β’ π΄ β V | |
2 | 1 | ssrankr1 9836 | . . 3 β’ (π΅ β On β (π΅ β (rankβπ΄) β Β¬ π΄ β (π 1βπ΅))) |
3 | rankon 9796 | . . . 4 β’ (rankβπ΄) β On | |
4 | ontri1 6398 | . . . 4 β’ ((π΅ β On β§ (rankβπ΄) β On) β (π΅ β (rankβπ΄) β Β¬ (rankβπ΄) β π΅)) | |
5 | 3, 4 | mpan2 688 | . . 3 β’ (π΅ β On β (π΅ β (rankβπ΄) β Β¬ (rankβπ΄) β π΅)) |
6 | 2, 5 | bitr3d 281 | . 2 β’ (π΅ β On β (Β¬ π΄ β (π 1βπ΅) β Β¬ (rankβπ΄) β π΅)) |
7 | 6 | con4bid 317 | 1 β’ (π΅ β On β (π΄ β (π 1βπ΅) β (rankβπ΄) β π΅)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β wcel 2105 Vcvv 3473 β wss 3948 Oncon0 6364 βcfv 6543 π 1cr1 9763 rankcrnk 9764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-reg 9593 ax-inf2 9642 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-r1 9765 df-rank 9766 |
This theorem is referenced by: r1val2 9838 r1pwALT 9847 elhf2 35619 gruex 43523 |
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