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Mirrors > Home > ILE Home > Th. List > 2oconcl | Unicode version |
Description: Closure of the pair
swapping function on ![]() |
Ref | Expression |
---|---|
2oconcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpri 3614 |
. . . . 5
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2 | difeq2 3247 |
. . . . . . . 8
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3 | dif0 3493 |
. . . . . . . 8
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4 | 2, 3 | eqtrdi 2226 |
. . . . . . 7
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5 | difeq2 3247 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | difid 3491 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | eqtrdi 2226 |
. . . . . . 7
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8 | 4, 7 | orim12i 759 |
. . . . . 6
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9 | 8 | orcomd 729 |
. . . . 5
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10 | 1, 9 | syl 14 |
. . . 4
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11 | 1on 6417 |
. . . . . 6
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12 | difexg 4141 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 11, 12 | ax-mp 5 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 13 | elpr 3612 |
. . . 4
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15 | 10, 14 | sylibr 134 |
. . 3
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16 | df2o3 6424 |
. . 3
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17 | 15, 16 | eleqtrrdi 2271 |
. 2
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18 | 17, 16 | eleq2s 2272 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-uni 3808 df-tr 4099 df-iord 4362 df-on 4364 df-suc 4367 df-1o 6410 df-2o 6411 |
This theorem is referenced by: ismkvnex 7146 |
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