![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > winalim | Structured version Visualization version GIF version |
Description: A weakly inaccessible cardinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
winalim | β’ (π΄ β Inaccw β Lim π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | winainf 10638 | . 2 β’ (π΄ β Inaccw β Ο β π΄) | |
2 | winacard 10636 | . . 3 β’ (π΄ β Inaccw β (cardβπ΄) = π΄) | |
3 | cardlim 9916 | . . . 4 β’ (Ο β (cardβπ΄) β Lim (cardβπ΄)) | |
4 | sseq2 3974 | . . . . 5 β’ ((cardβπ΄) = π΄ β (Ο β (cardβπ΄) β Ο β π΄)) | |
5 | limeq 6333 | . . . . 5 β’ ((cardβπ΄) = π΄ β (Lim (cardβπ΄) β Lim π΄)) | |
6 | 4, 5 | bibi12d 346 | . . . 4 β’ ((cardβπ΄) = π΄ β ((Ο β (cardβπ΄) β Lim (cardβπ΄)) β (Ο β π΄ β Lim π΄))) |
7 | 3, 6 | mpbii 232 | . . 3 β’ ((cardβπ΄) = π΄ β (Ο β π΄ β Lim π΄)) |
8 | 2, 7 | syl 17 | . 2 β’ (π΄ β Inaccw β (Ο β π΄ β Lim π΄)) |
9 | 1, 8 | mpbid 231 | 1 β’ (π΄ β Inaccw β Lim π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 β wss 3914 Lim wlim 6322 βcfv 6500 Οcom 7806 cardccrd 9879 Inaccwcwina 10626 |
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-sbc 3744 df-csb 3860 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-lim 6326 df-suc 6327 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-om 7807 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-card 9883 df-cf 9885 df-wina 10628 |
This theorem is referenced by: inar1 10719 inatsk 10722 tskuni 10727 grur1a 10763 |
Copyright terms: Public domain | W3C validator |