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Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > trirec0xor | Unicode version |
Description: Version of trirec0 13556 with exclusive-or.
The definition of a discrete field is sometimes stated in terms of exclusive-or but as proved here, this is equivalent to inclusive-or because the two disjuncts cannot be simultaneously true. (Contributed by Jim Kingdon, 10-Jun-2024.) |
Ref | Expression |
---|---|
trirec0xor |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trirec0 13556 |
. 2
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2 | 1ne0 8880 |
. . . . . . . 8
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3 | 2 | nesymi 2370 |
. . . . . . 7
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4 | simpr 109 |
. . . . . . . . . . 11
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5 | 4 | oveq1d 5829 |
. . . . . . . . . 10
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6 | mul02lem2 8242 |
. . . . . . . . . 10
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7 | 5, 6 | sylan9eqr 2209 |
. . . . . . . . 9
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8 | simprl 521 |
. . . . . . . . 9
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9 | 7, 8 | eqtr3d 2189 |
. . . . . . . 8
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10 | 9 | rexlimiva 2566 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 3, 10 | mto 652 |
. . . . . 6
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12 | r19.41v 2610 |
. . . . . 6
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13 | 11, 12 | mtbi 660 |
. . . . 5
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14 | 13 | biantru 300 |
. . . 4
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15 | df-xor 1355 |
. . . 4
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16 | 14, 15 | bitr4i 186 |
. . 3
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17 | 16 | ralbii 2460 |
. 2
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18 | 1, 17 | bitri 183 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-xor 1355 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-br 3962 df-opab 4022 df-id 4248 df-po 4251 df-iso 4252 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-iota 5128 df-fun 5165 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 |
This theorem is referenced by: (None) |
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