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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 7036 | . . . 4 | |
2 | 1 | a1i 9 | . . 3 |
3 | breq2 3933 | . . . . . 6 | |
4 | 3 | rabbidv 2675 | . . . . 5 |
5 | 4 | inteqd 3776 | . . . 4 |
6 | 5 | adantl 275 | . . 3 |
7 | encv 6640 | . . . . 5 | |
8 | 7 | simprd 113 | . . . 4 |
9 | 8 | rexlimivw 2545 | . . 3 |
10 | intexrabim 4078 | . . 3 | |
11 | 2, 6, 9, 10 | fvmptd 5502 | . 2 |
12 | onintrab2im 4434 | . 2 | |
13 | 11, 12 | eqeltrd 2216 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 wrex 2417 crab 2420 cvv 2686 cint 3771 class class class wbr 3929 cmpt 3989 con0 4285 cfv 5123 cen 6632 ccrd 7035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-en 6635 df-card 7036 |
This theorem is referenced by: (None) |
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