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Mirrors > Home > ILE Home > Th. List > domfiexmid | Unicode version |
Description: If any set dominated by a finite set is finite, excluded middle follows. (Contributed by Jim Kingdon, 3-Feb-2022.) |
Ref | Expression |
---|---|
domfiexmid.1 |
Ref | Expression |
---|---|
domfiexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3905 | . . . 4 | |
2 | snfig 6314 | . . . 4 | |
3 | 1, 2 | ax-mp 7 | . . 3 |
4 | ssrab2 3079 | . . . 4 | |
5 | ssdomg 6281 | . . . 4 | |
6 | 3, 4, 5 | mp2 16 | . . 3 |
7 | domfiexmid.1 | . . . . . 6 | |
8 | 7 | gen2 1379 | . . . . 5 |
9 | p0ex 3959 | . . . . . 6 | |
10 | eleq1 2141 | . . . . . . . . 9 | |
11 | breq2 3789 | . . . . . . . . 9 | |
12 | 10, 11 | anbi12d 456 | . . . . . . . 8 |
13 | 12 | imbi1d 229 | . . . . . . 7 |
14 | 13 | albidv 1745 | . . . . . 6 |
15 | 9, 14 | spcv 2691 | . . . . 5 |
16 | 8, 15 | ax-mp 7 | . . . 4 |
17 | 9 | rabex 3922 | . . . . 5 |
18 | breq1 3788 | . . . . . . 7 | |
19 | 18 | anbi2d 451 | . . . . . 6 |
20 | eleq1 2141 | . . . . . 6 | |
21 | 19, 20 | imbi12d 232 | . . . . 5 |
22 | 17, 21 | spcv 2691 | . . . 4 |
23 | 16, 22 | ax-mp 7 | . . 3 |
24 | 3, 6, 23 | mp2an 416 | . 2 |
25 | 24 | ssfilem 6360 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wo 661 wal 1282 wceq 1284 wcel 1433 crab 2352 cvv 2601 wss 2973 c0 3251 csn 3398 class class class wbr 3785 cdom 6243 cfn 6244 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1o 6024 df-er 6129 df-en 6245 df-dom 6246 df-fin 6247 |
This theorem is referenced by: (None) |
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