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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 10270 | . . 3 β’ (cfβπ΄) β (cardβπ΄) | |
2 | cardonle 9974 | . . 3 β’ (π΄ β On β (cardβπ΄) β π΄) | |
3 | 1, 2 | sstrid 3989 | . 2 β’ (π΄ β On β (cfβπ΄) β π΄) |
4 | cff 10265 | . . . . . 6 β’ cf:OnβΆOn | |
5 | 4 | fdmi 6728 | . . . . 5 β’ dom cf = On |
6 | 5 | eleq2i 2820 | . . . 4 β’ (π΄ β dom cf β π΄ β On) |
7 | ndmfv 6926 | . . . 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 4392 | . . 3 β’ β β π΄ | |
10 | 8, 9 | eqsstrdi 4032 | . 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 1534 β wcel 2099 β wss 3944 β c0 4318 dom cdm 5672 Oncon0 6363 βcfv 6542 cardccrd 9952 cfccf 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-en 8958 df-card 9956 df-cf 9958 |
This theorem is referenced by: cfom 10281 cfidm 10292 alephreg 10599 winafp 10714 tskcard 10798 gruina 10835 |
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