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Mirrors > Home > ILE Home > Th. List > cardcl | GIF version |
Description: The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
cardcl | β’ (βπ¦ β On π¦ β π΄ β (cardβπ΄) β On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-card 7181 | . . . 4 β’ card = (π₯ β V β¦ β© {π¦ β On β£ π¦ β π₯}) | |
2 | 1 | a1i 9 | . . 3 β’ (βπ¦ β On π¦ β π΄ β card = (π₯ β V β¦ β© {π¦ β On β£ π¦ β π₯})) |
3 | breq2 4009 | . . . . . 6 β’ (π₯ = π΄ β (π¦ β π₯ β π¦ β π΄)) | |
4 | 3 | rabbidv 2728 | . . . . 5 β’ (π₯ = π΄ β {π¦ β On β£ π¦ β π₯} = {π¦ β On β£ π¦ β π΄}) |
5 | 4 | inteqd 3851 | . . . 4 β’ (π₯ = π΄ β β© {π¦ β On β£ π¦ β π₯} = β© {π¦ β On β£ π¦ β π΄}) |
6 | 5 | adantl 277 | . . 3 β’ ((βπ¦ β On π¦ β π΄ β§ π₯ = π΄) β β© {π¦ β On β£ π¦ β π₯} = β© {π¦ β On β£ π¦ β π΄}) |
7 | encv 6748 | . . . . 5 β’ (π¦ β π΄ β (π¦ β V β§ π΄ β V)) | |
8 | 7 | simprd 114 | . . . 4 β’ (π¦ β π΄ β π΄ β V) |
9 | 8 | rexlimivw 2590 | . . 3 β’ (βπ¦ β On π¦ β π΄ β π΄ β V) |
10 | intexrabim 4155 | . . 3 β’ (βπ¦ β On π¦ β π΄ β β© {π¦ β On β£ π¦ β π΄} β V) | |
11 | 2, 6, 9, 10 | fvmptd 5599 | . 2 β’ (βπ¦ β On π¦ β π΄ β (cardβπ΄) = β© {π¦ β On β£ π¦ β π΄}) |
12 | onintrab2im 4519 | . 2 β’ (βπ¦ β On π¦ β π΄ β β© {π¦ β On β£ π¦ β π΄} β On) | |
13 | 11, 12 | eqeltrd 2254 | 1 β’ (βπ¦ β On π¦ β π΄ β (cardβπ΄) β On) |
Colors of variables: wff set class |
Syntax hints: β wi 4 = wceq 1353 β wcel 2148 βwrex 2456 {crab 2459 Vcvv 2739 β© cint 3846 class class class wbr 4005 β¦ cmpt 4066 Oncon0 4365 βcfv 5218 β cen 6740 cardccrd 7180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-en 6743 df-card 7181 |
This theorem is referenced by: (None) |
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