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Mirrors > Home > ILE Home > Th. List > unfiexmid | Unicode version |
Description: If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.) |
Ref | Expression |
---|---|
unfiexmid.1 |
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Ref | Expression |
---|---|
unfiexmid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3429 |
. . . . 5
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2 | unfiexmid.1 |
. . . . . . 7
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3 | 2 | rgen2a 2423 |
. . . . . 6
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4 | df1o2 6124 |
. . . . . . . . . 10
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5 | rabeq 2604 |
. . . . . . . . . 10
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6 | 4, 5 | ax-mp 7 |
. . . . . . . . 9
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7 | ordtriexmidlem 4298 |
. . . . . . . . 9
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8 | 6, 7 | eqeltri 2155 |
. . . . . . . 8
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9 | snfig 6459 |
. . . . . . . 8
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10 | 8, 9 | ax-mp 7 |
. . . . . . 7
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11 | 1onn 6207 |
. . . . . . . 8
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12 | snfig 6459 |
. . . . . . . 8
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13 | 11, 12 | ax-mp 7 |
. . . . . . 7
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14 | uneq1 3131 |
. . . . . . . . 9
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15 | 14 | eleq1d 2151 |
. . . . . . . 8
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16 | uneq2 3132 |
. . . . . . . . 9
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17 | 16 | eleq1d 2151 |
. . . . . . . 8
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18 | 15, 17 | rspc2v 2723 |
. . . . . . 7
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19 | 10, 13, 18 | mp2an 417 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 3, 19 | ax-mp 7 |
. . . . 5
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21 | 1, 20 | eqeltri 2155 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 8 | elexi 2622 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | prid1 3522 |
. . . 4
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24 | 11 | elexi 2622 |
. . . . 5
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25 | 24 | prid2 3523 |
. . . 4
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26 | fidceq 6513 |
. . . 4
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27 | 21, 23, 25, 26 | mp3an 1269 |
. . 3
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28 | exmiddc 778 |
. . 3
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29 | 27, 28 | ax-mp 7 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 4 | eqeq2i 2093 |
. . . 4
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31 | 0ex 3931 |
. . . . 5
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32 | biidd 170 |
. . . . 5
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33 | 31, 32 | rabsnt 3491 |
. . . 4
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34 | 30, 33 | sylbi 119 |
. . 3
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35 | iba 294 |
. . . . . 6
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36 | 35 | abbi2dv 2201 |
. . . . 5
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37 | df-rab 2362 |
. . . . 5
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38 | 36, 37 | syl6reqr 2134 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | 38 | con3i 595 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 34, 39 | orim12i 709 |
. 2
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41 | 29, 40 | ax-mp 7 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-tr 3902 df-id 4083 df-iord 4156 df-on 4158 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-1o 6111 df-en 6386 df-fin 6388 |
This theorem is referenced by: (None) |
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