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Mirrors > Home > ILE Home > Th. List > ssfiexmid | Unicode version |
Description: If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.) |
Ref | Expression |
---|---|
ssfiexmid.1 |
Ref | Expression |
---|---|
ssfiexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4025 | . . . 4 | |
2 | snfig 6676 | . . . 4 | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | ssrab2 3152 | . . 3 | |
5 | ssfiexmid.1 | . . . . 5 | |
6 | p0ex 4082 | . . . . . 6 | |
7 | eleq1 2180 | . . . . . . . . 9 | |
8 | sseq2 3091 | . . . . . . . . 9 | |
9 | 7, 8 | anbi12d 464 | . . . . . . . 8 |
10 | 9 | imbi1d 230 | . . . . . . 7 |
11 | 10 | albidv 1780 | . . . . . 6 |
12 | 6, 11 | spcv 2753 | . . . . 5 |
13 | 5, 12 | ax-mp 5 | . . . 4 |
14 | 6 | rabex 4042 | . . . . 5 |
15 | sseq1 3090 | . . . . . . 7 | |
16 | 15 | anbi2d 459 | . . . . . 6 |
17 | eleq1 2180 | . . . . . 6 | |
18 | 16, 17 | imbi12d 233 | . . . . 5 |
19 | 14, 18 | spcv 2753 | . . . 4 |
20 | 13, 19 | ax-mp 5 | . . 3 |
21 | 3, 4, 20 | mp2an 422 | . 2 |
22 | 21 | ssfilem 6737 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 682 wal 1314 wceq 1316 wcel 1465 crab 2397 cvv 2660 wss 3041 c0 3333 csn 3497 cfn 6602 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1o 6281 df-er 6397 df-en 6603 df-fin 6605 |
This theorem is referenced by: infiexmid 6739 |
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