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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 5866 |
. 2
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2 | ssun2 3107 |
. . . . 5
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3 | 0ex 3884 |
. . . . . 6
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4 | 3 | snid 3402 |
. . . . 5
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5 | 2, 4 | sselii 2942 |
. . . 4
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6 | 5 | biantrur 287 |
. . 3
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7 | cnvsn0 4789 |
. . . . . 6
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8 | 7 | unieqi 3590 |
. . . . 5
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9 | uni0 3607 |
. . . . 5
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10 | 8, 9 | eqtri 2060 |
. . . 4
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11 | 10 | breq1i 3771 |
. . 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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-tpos 5860 |
This theorem is referenced by: reldmtpos 5868 tpostpos 5879 |
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