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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 6847 | . . . . . 6 β’ (π¦ = π΄ β (rankβπ¦) = (rankβπ΄)) | |
2 | 1 | eleq1d 2823 | . . . . 5 β’ (π¦ = π΄ β ((rankβπ¦) β π₯ β (rankβπ΄) β π₯)) |
3 | 2 | ralsng 4639 | . . . 4 β’ (π΄ β βͺ (π 1 β On) β (βπ¦ β {π΄} (rankβπ¦) β π₯ β (rankβπ΄) β π₯)) |
4 | 3 | rabbidv 3418 | . . 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 4917 | . 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 9752 | . . 3 β’ (π΄ β βͺ (π 1 β On) β {π΄} β βͺ (π 1 β On)) | |
7 | rankval3b 9769 | . . 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 9738 | . . 3 β’ (rankβπ΄) β On | |
10 | onsucmin 7761 | . . 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 2787 | 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 1542 β wcel 2107 βwral 3065 {crab 3410 {csn 4591 βͺ cuni 4870 β© cint 4912 β cima 5641 Oncon0 6322 suc csuc 6324 βcfv 6501 π 1cr1 9705 rankcrnk 9706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-r1 9707 df-rank 9708 |
This theorem is referenced by: rankprb 9794 ranksn 9797 rankcf 10720 rankaltopb 34593 |
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