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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 6902 | . . . . . 6 β’ (π¦ = π΄ β (rankβπ¦) = (rankβπ΄)) | |
2 | 1 | eleq1d 2814 | . . . . 5 β’ (π¦ = π΄ β ((rankβπ¦) β π₯ β (rankβπ΄) β π₯)) |
3 | 2 | ralsng 4682 | . . . 4 β’ (π΄ β βͺ (π 1 β On) β (βπ¦ β {π΄} (rankβπ¦) β π₯ β (rankβπ΄) β π₯)) |
4 | 3 | rabbidv 3438 | . . 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 4958 | . 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 9840 | . . 3 β’ (π΄ β βͺ (π 1 β On) β {π΄} β βͺ (π 1 β On)) | |
7 | rankval3b 9857 | . . 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 9826 | . . 3 β’ (rankβπ΄) β On | |
10 | onsucmin 7830 | . . 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 2778 | 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 1533 β wcel 2098 βwral 3058 {crab 3430 {csn 4632 βͺ cuni 4912 β© cint 4953 β cima 5685 Oncon0 6374 suc csuc 6376 βcfv 6553 π 1cr1 9793 rankcrnk 9794 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-r1 9795 df-rank 9796 |
This theorem is referenced by: rankprb 9882 ranksn 9885 rankcf 10808 rankaltopb 35608 |
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