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Mirrors > Home > ILE Home > Th. List > pw1fin | Unicode version |
Description: Excluded middle is equivalent to the power set of being finite. (Contributed by SN and Jim Kingdon, 7-Aug-2024.) |
Ref | Expression |
---|---|
pw1fin | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidpweq 6851 | . . . 4 EXMID | |
2 | 1 | biimpi 119 | . . 3 EXMID |
3 | 2onn 6465 | . . . 4 | |
4 | nnfi 6814 | . . . 4 | |
5 | 3, 4 | ax-mp 5 | . . 3 |
6 | 2, 5 | eqeltrdi 2248 | . 2 EXMID |
7 | df1o2 6373 | . . . . . 6 | |
8 | 7 | sseq2i 3155 | . . . . 5 |
9 | velpw 3550 | . . . . . 6 | |
10 | 1oex 6368 | . . . . . . . 8 | |
11 | 10 | pwid 3558 | . . . . . . 7 |
12 | fidceq 6811 | . . . . . . 7 DECID | |
13 | 11, 12 | mp3an3 1308 | . . . . . 6 DECID |
14 | 9, 13 | sylan2br 286 | . . . . 5 DECID |
15 | 8, 14 | sylan2br 286 | . . . 4 DECID |
16 | 7 | eqeq2i 2168 | . . . . 5 |
17 | 16 | dcbii 826 | . . . 4 DECID DECID |
18 | 15, 17 | sylib 121 | . . 3 DECID |
19 | 18 | exmid1dc 4161 | . 2 EXMID |
20 | 6, 19 | impbii 125 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 DECID wdc 820 wceq 1335 wcel 2128 wss 3102 c0 3394 cpw 3543 csn 3560 EXMIDwem 4155 com 4548 c1o 6353 c2o 6354 cfn 6682 |
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 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 |
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 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-tr 4063 df-exmid 4156 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-1o 6360 df-2o 6361 df-en 6683 df-fin 6685 |
This theorem is referenced by: (None) |
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