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Mirrors > Home > ILE Home > Th. List > cardcl | Unicode version |
Description: The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Ref | Expression |
---|---|
cardcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-card 6360 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | breq2 3768 | . . . . . 6 | |
4 | 3 | rabbidv 2549 | . . . . 5 |
5 | 4 | inteqd 3620 | . . . 4 |
6 | 5 | adantl 262 | . . 3 |
7 | encv 6227 | . . . . 5 | |
8 | 7 | simprd 107 | . . . 4 |
9 | 8 | rexlimivw 2429 | . . 3 |
10 | intexrabim 3907 | . . 3 | |
11 | 2, 6, 9, 10 | fvmptd 5253 | . 2 |
12 | onintrab2im 4244 | . 2 | |
13 | 11, 12 | eqeltrd 2114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1243 wcel 1393 wrex 2307 crab 2310 cvv 2557 cint 3615 class class class wbr 3764 cmpt 3818 con0 4100 cfv 4902 cen 6219 ccrd 6359 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-en 6222 df-card 6360 |
This theorem is referenced by: (None) |
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