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Mirrors > Home > MPE Home > Th. List > cfle | Structured version Visualization version GIF version |
Description: Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
cfle | β’ (cfβπ΄) β π΄ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cflecard 10197 | . . 3 β’ (cfβπ΄) β (cardβπ΄) | |
2 | cardonle 9901 | . . 3 β’ (π΄ β On β (cardβπ΄) β π΄) | |
3 | 1, 2 | sstrid 3959 | . 2 β’ (π΄ β On β (cfβπ΄) β π΄) |
4 | cff 10192 | . . . . . 6 β’ cf:OnβΆOn | |
5 | 4 | fdmi 6684 | . . . . 5 β’ dom cf = On |
6 | 5 | eleq2i 2826 | . . . 4 β’ (π΄ β dom cf β π΄ β On) |
7 | ndmfv 6881 | . . . 4 β’ (Β¬ π΄ β dom cf β (cfβπ΄) = β ) | |
8 | 6, 7 | sylnbir 331 | . . 3 β’ (Β¬ π΄ β On β (cfβπ΄) = β ) |
9 | 0ss 4360 | . . 3 β’ β β π΄ | |
10 | 8, 9 | eqsstrdi 4002 | . 2 β’ (Β¬ π΄ β On β (cfβπ΄) β π΄) |
11 | 3, 10 | pm2.61i 182 | 1 β’ (cfβπ΄) β π΄ |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 β wss 3914 β c0 4286 dom cdm 5637 Oncon0 6321 βcfv 6500 cardccrd 9879 cfccf 9881 |
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 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-en 8890 df-card 9883 df-cf 9885 |
This theorem is referenced by: cfom 10208 cfidm 10219 alephreg 10526 winafp 10641 tskcard 10725 gruina 10762 |
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