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Mirrors > Home > MPE Home > Th. List > ranksnb | Structured version Visualization version GIF version |
Description: The rank of a singleton. Theorem 15.17(v) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) |
Ref | Expression |
---|---|
ranksnb | β’ (π΄ β βͺ (π 1 β On) β (rankβ{π΄}) = suc (rankβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6888 | . . . . . 6 β’ (π¦ = π΄ β (rankβπ¦) = (rankβπ΄)) | |
2 | 1 | eleq1d 2818 | . . . . 5 β’ (π¦ = π΄ β ((rankβπ¦) β π₯ β (rankβπ΄) β π₯)) |
3 | 2 | ralsng 4676 | . . . 4 β’ (π΄ β βͺ (π 1 β On) β (βπ¦ β {π΄} (rankβπ¦) β π₯ β (rankβπ΄) β π₯)) |
4 | 3 | rabbidv 3440 | . . 3 β’ (π΄ β βͺ (π 1 β On) β {π₯ β On β£ βπ¦ β {π΄} (rankβπ¦) β π₯} = {π₯ β On β£ (rankβπ΄) β π₯}) |
5 | 4 | inteqd 4954 | . 2 β’ (π΄ β βͺ (π 1 β On) β β© {π₯ β On β£ βπ¦ β {π΄} (rankβπ¦) β π₯} = β© {π₯ β On β£ (rankβπ΄) β π₯}) |
6 | snwf 9800 | . . 3 β’ (π΄ β βͺ (π 1 β On) β {π΄} β βͺ (π 1 β On)) | |
7 | rankval3b 9817 | . . 3 β’ ({π΄} β βͺ (π 1 β On) β (rankβ{π΄}) = β© {π₯ β On β£ βπ¦ β {π΄} (rankβπ¦) β π₯}) | |
8 | 6, 7 | syl 17 | . 2 β’ (π΄ β βͺ (π 1 β On) β (rankβ{π΄}) = β© {π₯ β On β£ βπ¦ β {π΄} (rankβπ¦) β π₯}) |
9 | rankon 9786 | . . 3 β’ (rankβπ΄) β On | |
10 | onsucmin 7805 | . . 3 β’ ((rankβπ΄) β On β suc (rankβπ΄) = β© {π₯ β On β£ (rankβπ΄) β π₯}) | |
11 | 9, 10 | mp1i 13 | . 2 β’ (π΄ β βͺ (π 1 β On) β suc (rankβπ΄) = β© {π₯ β On β£ (rankβπ΄) β π₯}) |
12 | 5, 8, 11 | 3eqtr4d 2782 | 1 β’ (π΄ β βͺ (π 1 β On) β (rankβ{π΄}) = suc (rankβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βwral 3061 {crab 3432 {csn 4627 βͺ cuni 4907 β© cint 4949 β cima 5678 Oncon0 6361 suc csuc 6363 βcfv 6540 π 1cr1 9753 rankcrnk 9754 |
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-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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-int 4950 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-r1 9755 df-rank 9756 |
This theorem is referenced by: rankprb 9842 ranksn 9845 rankcf 10768 rankaltopb 34939 |
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