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Mirrors > Home > MPE Home > Th. List > rankonid | Structured version Visualization version GIF version |
Description: The rank of an ordinal number is itself. Proposition 9.18 of [TakeutiZaring] p. 79 and its converse. (Contributed by NM, 14-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankonid | β’ (π΄ β dom π 1 β (rankβπ΄) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankonidlem 9818 | . . 3 β’ (π΄ β dom π 1 β (π΄ β βͺ (π 1 β On) β§ (rankβπ΄) = π΄)) | |
2 | 1 | simprd 495 | . 2 β’ (π΄ β dom π 1 β (rankβπ΄) = π΄) |
3 | id 22 | . . 3 β’ ((rankβπ΄) = π΄ β (rankβπ΄) = π΄) | |
4 | rankdmr1 9791 | . . 3 β’ (rankβπ΄) β dom π 1 | |
5 | 3, 4 | eqeltrrdi 2834 | . 2 β’ ((rankβπ΄) = π΄ β π΄ β dom π 1) |
6 | 2, 5 | impbii 208 | 1 β’ (π΄ β dom π 1 β (rankβπ΄) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1533 β wcel 2098 βͺ cuni 4899 dom cdm 5666 β cima 5669 Oncon0 6354 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-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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: rankeq0b 9850 rankr1id 9852 rankcf 10767 r1tskina 10772 rankeq1o 35604 hfninf 35619 |
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