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Mirrors > Home > ILE Home > Th. List > brtpos0 | Unicode version |
Description: The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
brtpos0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brtpos2 5807 |
. 2
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2 | ssun2 3101 |
. . . . 5
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3 | 0ex 3875 |
. . . . . 6
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4 | 3 | snid 3394 |
. . . . 5
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5 | 2, 4 | sselii 2936 |
. . . 4
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6 | 5 | biantrur 287 |
. . 3
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7 | cnvsn0 4732 |
. . . . . 6
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8 | 7 | unieqi 3581 |
. . . . 5
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9 | uni0 3598 |
. . . . 5
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10 | 8, 9 | eqtri 2057 |
. . . 4
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11 | 10 | breq1i 3762 |
. . 3
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12 | 6, 11 | bitr3i 175 |
. 2
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13 | 1, 12 | syl6bb 185 |
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 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 df-tpos 5801 |
This theorem is referenced by: reldmtpos 5809 tpostpos 5820 |
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