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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 6929 | . 2 DECID | |
2 | simpll 524 | . . . . . . 7 DECID DECID | |
3 | simpr 109 | . . . . . . 7 DECID | |
4 | simplr 525 | . . . . . . 7 DECID | |
5 | 2, 3, 4 | fidcenumlemr 6932 | . . . . . 6 DECID |
6 | 5 | ex 114 | . . . . 5 DECID |
7 | 6 | exlimdv 1812 | . . . 4 DECID |
8 | 7 | rexlimdva 2587 | . . 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 829 wex 1485 wcel 2141 wral 2448 wrex 2449 com 4574 wfo 5196 cfn 6718 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-1o 6395 df-er 6513 df-en 6719 df-fin 6721 |
This theorem is referenced by: finct 7093 ctinf 12385 |
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