![]() |
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 10717 | . 2 β’ (π΄ β Inaccw β Ο β π΄) | |
2 | winacard 10715 | . . 3 β’ (π΄ β Inaccw β (cardβπ΄) = π΄) | |
3 | cardlim 9995 | . . . 4 β’ (Ο β (cardβπ΄) β Lim (cardβπ΄)) | |
4 | sseq2 4006 | . . . . 5 β’ ((cardβπ΄) = π΄ β (Ο β (cardβπ΄) β Ο β π΄)) | |
5 | limeq 6381 | . . . . 5 β’ ((cardβπ΄) = π΄ β (Lim (cardβπ΄) β Lim π΄)) | |
6 | 4, 5 | bibi12d 345 | . . . 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 1534 β wcel 2099 β wss 3947 Lim wlim 6370 βcfv 6548 Οcom 7870 cardccrd 9958 Inaccwcwina 10705 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-om 7871 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-card 9962 df-cf 9964 df-wina 10707 |
This theorem is referenced by: inar1 10798 inatsk 10801 tskuni 10806 grur1a 10842 |
Copyright terms: Public domain | W3C validator |