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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 10630 | . 2 β’ (π΄ β Inaccw β (π΄ β β β§ (cfβπ΄) = π΄ β§ βπ₯ β π΄ βπ¦ β π΄ π₯ βΊ π¦)) | |
2 | cardcf 10196 | . . . 4 β’ (cardβ(cfβπ΄)) = (cfβπ΄) | |
3 | fveq2 6846 | . . . 4 β’ ((cfβπ΄) = π΄ β (cardβ(cfβπ΄)) = (cardβπ΄)) | |
4 | id 22 | . . . 4 β’ ((cfβπ΄) = π΄ β (cfβπ΄) = π΄) | |
5 | 2, 3, 4 | 3eqtr3a 2797 | . . 3 β’ ((cfβπ΄) = π΄ β (cardβπ΄) = π΄) |
6 | 5 | 3ad2ant2 1135 | . 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 1088 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 βwrex 3070 β c0 4286 class class class wbr 5109 βcfv 6500 βΊ csdm 8888 cardccrd 9879 cfccf 9881 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-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-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-er 8654 df-en 8890 df-card 9883 df-cf 9885 df-wina 10628 |
This theorem is referenced by: winalim 10639 winalim2 10640 gchina 10643 inar1 10719 |
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