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Mirrors > Home > MPE Home > Th. List > winacard | Structured version Visualization version GIF version |
Description: A weakly inaccessible cardinal is a cardinal. (Contributed by Mario Carneiro, 29-May-2014.) |
Ref | Expression |
---|---|
winacard | β’ (π΄ β Inaccw β (cardβπ΄) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elwina 10680 | . 2 β’ (π΄ β Inaccw β (π΄ β β β§ (cfβπ΄) = π΄ β§ βπ₯ β π΄ βπ¦ β π΄ π₯ βΊ π¦)) | |
2 | cardcf 10246 | . . . 4 β’ (cardβ(cfβπ΄)) = (cfβπ΄) | |
3 | fveq2 6884 | . . . 4 β’ ((cfβπ΄) = π΄ β (cardβ(cfβπ΄)) = (cardβπ΄)) | |
4 | id 22 | . . . 4 β’ ((cfβπ΄) = π΄ β (cfβπ΄) = π΄) | |
5 | 2, 3, 4 | 3eqtr3a 2790 | . . 3 β’ ((cfβπ΄) = π΄ β (cardβπ΄) = π΄) |
6 | 5 | 3ad2ant2 1131 | . 2 β’ ((π΄ β β β§ (cfβπ΄) = π΄ β§ βπ₯ β π΄ βπ¦ β π΄ π₯ βΊ π¦) β (cardβπ΄) = π΄) |
7 | 1, 6 | sylbi 216 | 1 β’ (π΄ β Inaccw β (cardβπ΄) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 βwrex 3064 β c0 4317 class class class wbr 5141 βcfv 6536 βΊ csdm 8937 cardccrd 9929 cfccf 9931 Inaccwcwina 10676 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-er 8702 df-en 8939 df-card 9933 df-cf 9935 df-wina 10678 |
This theorem is referenced by: winalim 10689 winalim2 10690 gchina 10693 inar1 10769 |
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