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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 6016 |
. 2
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2 | ssun2 3164 |
. . . . 5
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3 | 0ex 3966 |
. . . . . 6
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4 | 3 | snid 3475 |
. . . . 5
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5 | 2, 4 | sselii 3022 |
. . . 4
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6 | 5 | biantrur 297 |
. . 3
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7 | cnvsn0 4899 |
. . . . . 6
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8 | 7 | unieqi 3663 |
. . . . 5
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9 | uni0 3680 |
. . . . 5
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10 | 8, 9 | eqtri 2108 |
. . . 4
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11 | 10 | breq1i 3852 |
. . 3
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12 | 6, 11 | bitr3i 184 |
. 2
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13 | 1, 12 | syl6bb 194 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-mpt 3901 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-fv 5023 df-tpos 6010 |
This theorem is referenced by: reldmtpos 6018 tpostpos 6029 |
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