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Mirrors > Home > MPE Home > Th. List > ssrankr1 | Structured version Visualization version GIF version |
Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets π 1. Proposition 9.15(3) of [TakeutiZaring] p. 79. (Contributed by NM, 8-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankid.1 | β’ π΄ β V |
Ref | Expression |
---|---|
ssrankr1 | β’ (π΅ β On β (π΅ β (rankβπ΄) β Β¬ π΄ β (π 1βπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankid.1 | . . . 4 β’ π΄ β V | |
2 | unir1 9803 | . . . 4 β’ βͺ (π 1 β On) = V | |
3 | 1, 2 | eleqtrri 2824 | . . 3 β’ π΄ β βͺ (π 1 β On) |
4 | r1fnon 9757 | . . . . . 6 β’ π 1 Fn On | |
5 | fndm 6642 | . . . . . 6 β’ (π 1 Fn On β dom π 1 = On) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 β’ dom π 1 = On |
7 | 6 | eleq2i 2817 | . . . 4 β’ (π΅ β dom π 1 β π΅ β On) |
8 | 7 | biimpri 227 | . . 3 β’ (π΅ β On β π΅ β dom π 1) |
9 | rankr1clem 9810 | . . 3 β’ ((π΄ β βͺ (π 1 β On) β§ π΅ β dom π 1) β (Β¬ π΄ β (π 1βπ΅) β π΅ β (rankβπ΄))) | |
10 | 3, 8, 9 | sylancr 586 | . 2 β’ (π΅ β On β (Β¬ π΄ β (π 1βπ΅) β π΅ β (rankβπ΄))) |
11 | 10 | bicomd 222 | 1 β’ (π΅ β On β (π΅ β (rankβπ΄) β Β¬ π΄ β (π 1βπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 = wceq 1533 β wcel 2098 Vcvv 3466 β wss 3940 βͺ cuni 4899 dom cdm 5666 β cima 5669 Oncon0 6354 Fn wfn 6528 βcfv 6533 π 1cr1 9752 rankcrnk 9753 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-reg 9582 ax-inf2 9631 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-r1 9754 df-rank 9755 |
This theorem is referenced by: rankr1a 9826 |
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