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Mirrors > Home > MPE Home > Th. List > isnum2 | Structured version Visualization version GIF version |
Description: A way to express well-orderability without bound or distinct variables. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
isnum2 | β’ (π΄ β dom card β βπ₯ β On π₯ β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardf2 9935 | . . . 4 β’ card:{π¦ β£ βπ₯ β On π₯ β π¦}βΆOn | |
2 | 1 | fdmi 6727 | . . 3 β’ dom card = {π¦ β£ βπ₯ β On π₯ β π¦} |
3 | 2 | eleq2i 2826 | . 2 β’ (π΄ β dom card β π΄ β {π¦ β£ βπ₯ β On π₯ β π¦}) |
4 | relen 8941 | . . . . 5 β’ Rel β | |
5 | 4 | brrelex2i 5732 | . . . 4 β’ (π₯ β π΄ β π΄ β V) |
6 | 5 | rexlimivw 3152 | . . 3 β’ (βπ₯ β On π₯ β π΄ β π΄ β V) |
7 | breq2 5152 | . . . 4 β’ (π¦ = π΄ β (π₯ β π¦ β π₯ β π΄)) | |
8 | 7 | rexbidv 3179 | . . 3 β’ (π¦ = π΄ β (βπ₯ β On π₯ β π¦ β βπ₯ β On π₯ β π΄)) |
9 | 6, 8 | elab3 3676 | . 2 β’ (π΄ β {π¦ β£ βπ₯ β On π₯ β π¦} β βπ₯ β On π₯ β π΄) |
10 | 3, 9 | bitri 275 | 1 β’ (π΄ β dom card β βπ₯ β On π₯ β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1542 β wcel 2107 {cab 2710 βwrex 3071 Vcvv 3475 class class class wbr 5148 dom cdm 5676 Oncon0 6362 β cen 8933 cardccrd 9927 |
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 5299 ax-nul 5306 ax-pr 5427 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6365 df-on 6366 df-fun 6543 df-fn 6544 df-f 6545 df-en 8937 df-card 9931 |
This theorem is referenced by: isnumi 9938 ennum 9939 xpnum 9943 cardval3 9944 dfac10c 10130 isfin7-2 10388 numth2 10463 inawinalem 10681 |
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