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Mirrors > Home > ILE Home > Th. List > fidcenum | Unicode version |
Description: A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.) |
Ref | Expression |
---|---|
fidcenum | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidcenumlemim 6899 | . 2 DECID | |
2 | simpll 519 | . . . . . . 7 DECID DECID | |
3 | simpr 109 | . . . . . . 7 DECID | |
4 | simplr 520 | . . . . . . 7 DECID | |
5 | 2, 3, 4 | fidcenumlemr 6902 | . . . . . 6 DECID |
6 | 5 | ex 114 | . . . . 5 DECID |
7 | 6 | exlimdv 1799 | . . . 4 DECID |
8 | 7 | rexlimdva 2574 | . . 3 DECID |
9 | 8 | imp 123 | . 2 DECID |
10 | 1, 9 | impbii 125 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 DECID wdc 820 wex 1472 wcel 2128 wral 2435 wrex 2436 com 4552 wfo 5171 cfn 6688 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4029 df-tr 4066 df-id 4256 df-iord 4329 df-on 4331 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-1o 6366 df-er 6483 df-en 6689 df-fin 6691 |
This theorem is referenced by: finct 7063 ctinf 12255 |
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