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Mirrors > Home > ILE Home > Th. List > exmidsssn | Unicode version |
Description: Excluded middle is equivalent to the biconditionalized version of sssnr 3765 for sets. (Contributed by Jim Kingdon, 5-Mar-2023.) |
Ref | Expression |
---|---|
exmidsssn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3473 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() | |
2 | sseq1 3190 |
. . . . . . 7
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3 | 1, 2 | mpbiri 168 |
. . . . . 6
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4 | 3 | adantl 277 |
. . . . 5
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5 | simpr 110 |
. . . . . 6
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6 | 5 | orcd 734 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | 4, 6 | 2thd 175 |
. . . 4
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8 | sssnm 3766 |
. . . . . 6
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9 | neq0r 3449 |
. . . . . . 7
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10 | biorf 745 |
. . . . . . 7
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11 | 9, 10 | syl 14 |
. . . . . 6
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12 | 8, 11 | bitrd 188 |
. . . . 5
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13 | 12 | adantl 277 |
. . . 4
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14 | exmidn0m 4213 |
. . . . . 6
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15 | 14 | biimpi 120 |
. . . . 5
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16 | 15 | 19.21bi 1568 |
. . . 4
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17 | 7, 13, 16 | mpjaodan 799 |
. . 3
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18 | 17 | alrimivv 1885 |
. 2
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19 | 0ex 4142 |
. . . . . 6
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20 | sneq 3615 |
. . . . . . . 8
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21 | 20 | sseq2d 3197 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 20 | eqeq2d 2199 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | orbi2d 791 |
. . . . . . 7
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24 | 21, 23 | bibi12d 235 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 19, 24 | spcv 2843 |
. . . . 5
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26 | 25 | biimpd 144 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | alimi 1465 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | exmid01 4210 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 27, 28 | sylibr 134 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 18, 29 | impbii 126 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-rab 2474 df-v 2751 df-dif 3143 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-exmid 4207 |
This theorem is referenced by: exmidsssnc 4215 |
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