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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 10686 | . 2 β’ (π΄ β Inaccw β Ο β π΄) | |
2 | winacard 10684 | . . 3 β’ (π΄ β Inaccw β (cardβπ΄) = π΄) | |
3 | cardlim 9964 | . . . 4 β’ (Ο β (cardβπ΄) β Lim (cardβπ΄)) | |
4 | sseq2 4001 | . . . . 5 β’ ((cardβπ΄) = π΄ β (Ο β (cardβπ΄) β Ο β π΄)) | |
5 | limeq 6367 | . . . . 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 1533 β wcel 2098 β wss 3941 Lim wlim 6356 βcfv 6534 Οcom 7849 cardccrd 9927 Inaccwcwina 10674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-om 7850 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-card 9931 df-cf 9933 df-wina 10676 |
This theorem is referenced by: inar1 10767 inatsk 10770 tskuni 10775 grur1a 10811 |
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