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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 10710 | . 2 β’ (π΄ β Inaccw β (π΄ β β β§ (cfβπ΄) = π΄ β§ βπ₯ β π΄ βπ¦ β π΄ π₯ βΊ π¦)) | |
2 | cardcf 10276 | . . . 4 β’ (cardβ(cfβπ΄)) = (cfβπ΄) | |
3 | fveq2 6897 | . . . 4 β’ ((cfβπ΄) = π΄ β (cardβ(cfβπ΄)) = (cardβπ΄)) | |
4 | id 22 | . . . 4 β’ ((cfβπ΄) = π΄ β (cfβπ΄) = π΄) | |
5 | 2, 3, 4 | 3eqtr3a 2792 | . . 3 β’ ((cfβπ΄) = π΄ β (cardβπ΄) = π΄) |
6 | 5 | 3ad2ant2 1132 | . 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 1085 = wceq 1534 β wcel 2099 β wne 2937 βwral 3058 βwrex 3067 β c0 4323 class class class wbr 5148 βcfv 6548 βΊ csdm 8963 cardccrd 9959 cfccf 9961 Inaccwcwina 10706 |
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-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-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-er 8725 df-en 8965 df-card 9963 df-cf 9965 df-wina 10708 |
This theorem is referenced by: winalim 10719 winalim2 10720 gchina 10723 inar1 10799 |
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